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A215078
Triangle of sums of the first k n-th powers multiplied by binomial(n,k), read by rows.
3
0, 0, 1, 0, 2, 5, 0, 3, 27, 36, 0, 4, 102, 392, 354, 0, 5, 330, 2760, 6500, 4425, 0, 6, 975, 15880, 73350, 123090, 67171, 0, 7, 2709, 81060, 654500, 2033325, 2637327, 1200304, 0, 8, 7196, 381808, 5064780, 25926824, 59992660, 63259168, 24684612, 0, 9, 18468, 1696464, 35574840, 281668590, 1034305524, 1896003648, 1681960464, 574304985, 0, 10, 46125, 7208880, 232816500, 2740317300, 14981494710, 42457884000, 64240088580, 49143419250, 14914341925
OFFSET
0,5
COMMENTS
If one starts the sum at j=0, the initial term T(0,0) is 1.
LINKS
FORMULA
T(n,k) = binomial(n,k)*sum(j^n, j=1..k)
EXAMPLE
0
0 1
0 2 5
0 3 27 36
0 4 102 392 354
0 5 330 2760 6500 4425
0 6 975 15880 73350 123090 67171
0 7 2709 81060 654500 2033325 2637327 1200304
MAPLE
A215078 := proc(n, k)
binomial(n, k)*add(j^n, j=1..k) ;
end proc:
seq(seq(A215078(n, k), k=0..n), n=0..10) ; # R. J. Mathar, Jan 27 2023
MATHEMATICA
Flatten[Table[Table[Sum[j^n, {j, 1, k}]*Binomial[n, k], {k, 0, n}], {n, 0, 10}], 1]
CROSSREFS
Binomial convolution of A215083.
Cf. A215077 (row sums), A031971 (diagonal)
Sequence in context: A208476 A247449 A112695 * A067881 A307649 A024714
KEYWORD
nonn,tabl
AUTHOR
Olivier Gérard, Aug 02 2012
STATUS
approved