A304330 T(n, k) = Sum_{j=0..k} (-1)^j*binomial(2*k, j)*(k - j)^(2*n), triangle read by rows, n >= 0 and 0 <= k <= n.

1, 0, 1, 0, 1, 12, 0, 1, 60, 360, 0, 1, 252, 5040, 20160, 0, 1, 1020, 52920, 604800, 1814400, 0, 1, 4092, 506880, 12640320, 99792000, 239500800, 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600, 0, 1, 65532, 42653520, 3952428480

Offset

0,6

Links

Table of n, a(n) for n=0..40.

José L. Cereceda, Sums of powers of integers and the sequence A304330, arXiv:2405.05268 [math.GM], 2024.

Example

Triangle starts:
[0] 1;
[1] 0, 1;
[2] 0, 1,    12;
[3] 0, 1,    60,     360;
[4] 0, 1,   252,    5040,     20160;
[5] 0, 1,  1020,   52920,    604800,    1814400;
[6] 0, 1,  4092,  506880,  12640320,   99792000,   239500800;
[7] 0, 1, 16380, 4684680, 230630400, 3632428800, 21794572800, 43589145600;

Maple

T := (n, k) -> add((-1)^j*binomial(2*k, j)*(k-j)^(2*n), j=0..k):
for n from 0 to 8 do seq(T(n, k), k=0..n) od;

Crossrefs

Row sums are A100872, T(n,2) = A058896, T(n,n) = A002674, T(n,n-1)= A091032.

Cf. A304334, A304336.

Sequence in context: A156401 A246223 A199542 * A322731 A370330 A370430

Adjacent sequences: A304327 A304328 A304329 * A304331 A304332 A304333

Keyword

nonn,tabl

Author

Peter Luschny, May 11 2018

Status

approved