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A080852
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Square array of 4D pyramidal numbers, read by antidiagonals.
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16
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1, 1, 4, 1, 5, 10, 1, 6, 15, 20, 1, 7, 20, 35, 35, 1, 8, 25, 50, 70, 56, 1, 9, 30, 65, 105, 126, 84, 1, 10, 35, 80, 140, 196, 210, 120, 1, 11, 40, 95, 175, 266, 336, 330, 165, 1, 12, 45, 110, 210, 336, 462, 540, 495, 220, 1, 13, 50, 125, 245, 406, 588, 750, 825, 715, 286, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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The first row contains the tetrahedral numbers, which are really three-dimensional, but can be regarded as degenerate 4D pyramidal numbers. - N. J. A. Sloane, Aug 28 2015
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LINKS
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Table of n, a(n) for n=0..66.
Index to sequences related to pyramidal numbers
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FORMULA
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T(n, k) = binomial(k + 4, 4) + (n-1)*binomial(k + 3, 4), corrected Oct 01 2021.
T(n, k) = T(n - 1, k) + C(k + 3, 4) = T(n - 1, k) + k(k + 1)(k + 2)(k + 3)/24.
G.f. for rows: (1 + nx)/(1 - x)^5, n >= -1.
T(n,k) = sum_{j=0..k} A080851(n,j). - R. J. Mathar, Jul 28 2016
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EXAMPLE
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Array, n >= 0, k >= 0, begins
1 4 10 20 35 56 ...
1 5 15 35 70 126 ...
1 6 20 50 105 196 ...
1 7 25 65 140 266 ...
1 8 30 80 175 336 ...
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MAPLE
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A080852 := proc(n, k)
binomial(k+4, 4)+(n-1)*binomial(k+3, 4) ;
end proc:
seq( seq(A080852(d-k, k), k=0..d), d=0..12) ; # R. J. Mathar, Oct 01 2021
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PROG
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(Derive) vector(vector(poly_coeff(Taylor((1+kx)/(1-x)^5, x, 11), x, n), n, 0, 11), k, -1, 10)
(Derive) VECTOR(VECTOR(comb(k+3, 3)+comb(k+3, 4)n, k, 0, 11), n, 0, 11)
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CROSSREFS
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Rows include A000292, A000332, A002415, A001296, A002418, A002419, A051740, A051797.
Cf. A057145, A080851, A180266, A055796 (antidiagonal sums).
See A257200 for another version of the array.
Sequence in context: A155060 A153426 A261720 * A204201 A090842 A120868
Adjacent sequences: A080849 A080850 A080851 * A080853 A080854 A080855
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Paul Barry, Feb 21 2003
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STATUS
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approved
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