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A215077
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Binomial convolution of sum of consecutive powers
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4
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0, 1, 7, 66, 852, 14020, 280472, 6609232, 179317056, 5505532992, 188717617280, 7143999854464, 296013377405440, 13325516967972352, 647610246703508480, 33794224057227356160, 1884620857353101983744, 111857608180484932648960, 7040178644779119413723136, 468349192560992552808841216, 32836927387372039917034405888
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OFFSET
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0,3
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COMMENTS
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a(0) could alternatively be defined as 1 from the formula or the convention for 0^0.
This sum is remarkable for its three different decompositions involving powers and binomials (see formulas and cross-refs)
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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a(n)=sum( binomial(n,k)*sum( j^n, j=1..k), k=0..n)
a(n)=sum( binomial(n,k)*H_k^{-n}, k=0..n) with H_k^(-n) = k-th harmonic number of order -n.
a(n)=sum( k^n * sum( binomial(n,n-k-j), j=0..n-k), k=0..n)
a(n)=sum( k^n * binomial(n,n-k) 2F1(1, k-n; k+1)(-1), k=0..n)
a(n)=sum( sum( (k-j)^n * binomial(n,j), j=0..k), k=0..n)
a(n)=sum( sum( (n-j)^n * binomial(n,n+k-j), j=0..n), k=0..n)
and the equivalent formulas obtained by symmetries of the binomial and the hypergeometric function as well as treating the zero-th term separately.
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MATHEMATICA
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Table[Sum[Sum[j^n*Binomial[n, k], {j, 1, k}], {k, 0, n}], {n, 0, 20}]
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PROG
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(PARI) a(n)=sum(k=0, n, binomial(n, k)*sum(j=1, k, j^n)) \\ Charles R Greathouse IV, Jul 31 2016
(PARI) a(n)=my(P=sumformal('x^n)); sum(k=0, n, binomial(n, k)*subst(P, 'x, k)) \\ Charles R Greathouse IV, Jul 31 2016
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CROSSREFS
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Row sums of A215078, A215079, A215080.
See also A215083 and A215084.
Sequence in context: A122705 A185181 A024395 * A003286 A244602 A223889
Adjacent sequences: A215074 A215075 A215076 * A215078 A215079 A215080
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KEYWORD
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nonn,nice
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AUTHOR
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Olivier Gérard, Aug 02 2012
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STATUS
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approved
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