0,13

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

Table of n, a(n) for n=0..77.

Conjecture: The columns are palindromes with sums A298311.

Triangle begins:

  1

  0  1

  0  1  1

  0  1  1  1

  0  0  3  1  1

  0  0  1  4  1  1

  0  0  1  4  4  1  1

  0  0  0  4  5  4  1  1

  0  0  0  1 10  5  4  1  1

  0  0  0  1  5 13  5  4  1  1

  0  0  0  0  4 13 14  5  4  1  1

  0  0  0  0  1 13 17 14  5  4  1  1

  0  0  0  0  1  5 28 18 14  5  4  1  1

Row n = 7 counts the following partitions:

  .  .  .  (322)      (43)      (52)     (61)  (7)

           (331)      (421)     (511)

           (2221)     (3211)    (4111)

           (1111111)  (22111)   (31111)

                      (211111)

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];

Table[Length[Select[IntegerPartitions[n], skats[#]==k&]], {n, 0, 12}, {k, -n, n, 2}]

Row sums are A000041.

Number of nonzero entries in row n appears to be A004396(n+1).

First nonzero entry of each row appears to converge to A146325.

The central column is A035544, half A035363.

Column sums appear to be A298311.

For original alternating sum we have A344651, ordered A097805.

The half-alternating version is A357637.

The ordered version (compositions) is A357646, half A357645.

The reverse version is A357705, half A357704.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357621 gives half-alternating sum of standard compositions, skew A357623.

A357629 gives half-alternating sum of prime indices, skew A357630.

A357633 gives half-alternating sum of Heinz partition, skew  A357634.

Cf. A035594, A053251, A357136, A357189, A357486, A357487, A357488, A357624, A357631, A357632, A357636.

nonn,tabl,new

Gus Wiseman, Oct 10 2022

approved

0,6

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

Table of n, a(n) for n=0..90.

Triangle begins:

  1

  0  1

  0  0  2

  0  0  1  2

  0  0  2  0  3

  0  0  2  2  0  3

  0  0  3  1  3  0  4

  0  0  3  2  4  2  0  4

  0  0  4  2  6  2  3  0  5

  0  0  4  3  5  7  3  3  0  5

  0  0  5  3  8  4 10  2  4  0  6

  0  0  5  4  8  6 11  9  3  4  0  6

  0  0  6  4 11  5 15  8 13  3  5  0  7

  0  0  6  5 11  8 13 19 10 13  4  5  0  7

  0  0  7  5 14  8 19 13 25  9 17  4  6  0  8

  0  0  7  6 14 11 19 17 29 23 13 18  5  6  0  8

Row n = 7 counts the following reversed partitions:

  .  .  (115)   (124)   (133)      (11113)   .  (7)

        (1114)  (1222)  (223)      (111112)     (16)

        (1123)          (11122)                 (25)

                        (1111111)               (34)

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];

Table[Length[Select[Reverse/@IntegerPartitions[n], halfats[#]==k&]], {n, 0, 15}, {k, -n, n, 2}]

Row sums are A000041.

Last entry of row n is A008619(n).

The central column in the non-reverse case is A035363, skew A035544.

For original reverse-alternating sum we have A344612.

For original alternating sum we have A344651, ordered A097805.

The non-reverse version is A357637, skew A357638.

The central column is A357639, skew A357640.

The non-reverse ordered version (compositions) is A357645, skew A357646.

The skew-alternating version is A357705.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357621 gives half-alternating sum of standard compositions, skew A357623.

A357629 gives half-alternating sum of prime indices, skew A357630.

A357633 gives half-alternating sum of Heinz partition, skew  A357634.

Cf. A029862, A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.

nonn,tabl,new

Gus Wiseman, Oct 10 2022

approved

0,8

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ...

Table of n, a(n) for n=0..77.

Triangle begins:

  1

  0  1

  0  1  1

  0  2  0  1

  0  2  2  0  1

  0  3  1  2  0  1

  0  3  2  3  2  0  1

  0  4  2  4  1  3  0  1

  0  4  3  3  6  2  3  0  1

  0  5  3  5  3  7  2  4  0  1

  0  5  4  5  4  9  7  3  4  0  1

  0  6  4  7  3 12  5 10  3  5  0  1

  0  6  5  7  5 10 16  7 11  4  5  0  1

  0  7  5  9  5 14 11 18  7 14  4  6  0  1

Row n = 7 counts the following reversed partitions:

  .  (16)   (25)   (34)       (1123)  (1114)   .  (7)

     (115)  (223)  (1222)             (11113)

     (124)         (111112)           (11122)

     (133)         (1111111)

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];

Table[Length[Select[Reverse/@IntegerPartitions[n], skats[#]==k&]], {n, 0, 11}, {k, -n, n, 2}]

Row sums are A000041.

First nonzero entry of each row is A004526.

The central column is A357640, half A357639.

For original alternating sum we have A344651, ordered A097805.

The half-alternating version is A357704.

The ordered non-reverse version (compositions) is A357646, half A357645.

The non-reverse version is A357638, half A357637.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357621 gives half-alternating sum of standard compositions, skew A357623.

A357629 gives half-alternating sum of prime indices, skew A357630.

A357633 gives half-alternating sum of Heinz partition, skew  A357634.

Cf. A035363, A035594, A053251, A298311, A357136, A357189, A357487, A357488, A357624, A357631, A357632, A357636.

nonn,tabl,new

Gus Wiseman, Oct 10 2022

approved

1,2

We define the skew-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A - B - C + D + E - F - G + ....

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Table of n, a(n) for n=1..55.

The terms together with their prime indices begin:

    1: {}

    4: {1,1}

    9: {2,2}

   12: {1,1,2}

   16: {1,1,1,1}

   25: {3,3}

   30: {1,2,3}

   36: {1,1,2,2}

   49: {4,4}

   63: {2,2,4}

   64: {1,1,1,1,1,1}

   70: {1,3,4}

   81: {2,2,2,2}

   90: {1,2,2,3}

  100: {1,1,3,3}

  108: {1,1,2,2,2}

  121: {5,5}

  144: {1,1,1,1,2,2}

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];

Select[Range[1000], skats[Reverse[primeMS[#]]]==0&]

The version for original alternating sum is A000290.

The half-alternating form is A000583, non-reverse A357631.

The version for standard compositions is A357628, non-reverse A357627.

The non-reverse version is A357632.

Positions of zeros in A357634, non-reverse A357630.

These partitions are counted by A357640, half A357639.

A056239 adds up prime indices, row sums of A112798.

A316524 gives alternating sum of prime indices, reverse A344616.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357641 counts comps w/ half-alt sum 0, even A357642.

Cf. A003963, A035594, A053251, A055932, A357189, A357485-A357488, A357621-A357626, A357629, A357637, A357638.

nonn,new

Gus Wiseman, Oct 09 2022

approved

1,12

Table of n, a(n) for n=1..56.

Cf. A000104, A144553.

nonn,new

John Mason, Oct 10 2022

approved

0,6

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

Table of n, a(n) for n=0..77.

Conjecture: The column sums are A029862.

Triangle begins:

  1

  0  1

  0  0  2

  0  0  1  2

  0  0  1  1  3

  0  0  0  2  2  3

  0  0  0  0  5  2  4

  0  0  0  0  2  6  3  4

  0  0  0  0  2  3  9  3  5

  0  0  0  0  0  4  7 10  4  5

  0  0  0  0  0  0 11  8 13  4  6

  0  0  0  0  0  0  4 15 12 14  5  6

  0  0  0  0  0  0  3  7 25 13 17  5  7

Row n = 9 counts the following partitions:

  (3222)       (333)      (432)     (441)  (9)

  (22221)      (3321)     (522)     (531)  (54)

  (21111111)   (4221)     (4311)    (621)  (63)

  (111111111)  (32211)    (5211)    (711)  (72)

               (222111)   (6111)           (81)

               (2211111)  (33111)

               (3111111)  (42111)

                          (51111)

                          (321111)

                          (411111)

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];

Table[Length[Select[IntegerPartitions[n], halfats[#]==k&]], {n, 0, 12}, {k, -n, n, 2}]

Row sums are A000041.

Number of nonzero entries in row n appears to be A004525(n+1).

Last entry of row n is A008619(n).

Column sums appear to be A029862.

The central column is A035363, skew A035544.

For original alternating sum we have A344651, ordered A097805.

The skew-alternating version is A357638.

The central column of the reverse is A357639, skew A357640.

The ordered version (compositions) is A357645, skew A357646.

The reverse version is A357704, skew A357705.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357621 gives  half-alternating sum of standard compositions, skew A357623.

A357629 gives half-alternating sum of prime indices, skew A357630.

A357633 gives half-alternating sum of Heinz partition, skew  A357634.

Cf. A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.

nonn,tabl,new

Gus Wiseman, Oct 10 2022

approved

9,2

Table of n, a(n) for n=9..47.

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^9.

a(n) = A121548(n,9).

nmax = 47; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^9, {x, 0, nmax}], x] // Drop[#, 9] &

Cf. A000045, A076739, A121548, A121549, A121550, A319402, A357688, A357690, A357691, A357694, A357716.

nonn,new

Ilya Gutkovskiy, Oct 10 2022

approved

8,2

Table of n, a(n) for n=8..48.

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^8.

a(n) = A121548(n,8).

nmax = 48; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^8, {x, 0, nmax}], x] // Drop[#, 8] &

Cf. A000045, A076739, A121548, A121549, A121550, A319401, A357688, A357690, A357691, A357694, A357717.

nonn,new

Ilya Gutkovskiy, Oct 10 2022

approved

7,2

Table of n, a(n) for n=7..49.

G.f.: ( Sum_{k>=2} x^Fibonacci(k) )^7.

a(n) = A121548(n,7).

nmax = 49; CoefficientList[Series[Sum[x^Fibonacci[k], {k, 2, 21}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

Cf. A000045, A076739, A121548, A121549, A121550, A319400, A357688, A357690, A357691, A357716, A357717.

nonn,new

Ilya Gutkovskiy, Oct 10 2022

approved

1,1

Although this is twice A143850, it is important enough to warrant an entry of it own. - N. J. A. Sloane, Oct 10 2022

Table of n, a(n) for n=1..51.

Cf. A045636, A103739, A143850, A227697.

nonn,new

Giuseppe Melfi, Oct 06 2022

approved