A382289 Irregular triangle T(n,k), n >= 0, 0 <= k <= 2*n+1, read by rows, where T(n,k) = [x^k] (1-x)^(n+1) * Sum_{k=0..n} (2*k+1)^n * x^k.

1, -1, 1, 1, -5, 3, 1, 6, 1, -49, 66, -25, 1, 23, 23, 1, -729, 1585, -1247, 343, 1, 76, 230, 76, 1, -14641, 44644, -54230, 30404, -6561, 1, 237, 1682, 1682, 237, 1, -371293, 1468383, -2433002, 2078278, -907257, 161051, 1, 722, 10543, 23548, 10543, 722, 1, -11390625, 55596806, -117286023, 135337972, -89493503, 32016102, -4826809

Offset

0,5

Links

Table of n, a(n) for n=0..55.

Formula

T(n,k) = A060187(n+1,k+1) for 0 <= k <= n.

T(n,k) = Sum_{j=0..n} (-1)^(k-j) * (2*j+1)^n * binomial(n+1,k-j).

Example

Irregular triangle begins:
  1, -1;
  1,  1,  -5,   3;
  1,  6,   1, -49,   66,    -25;
  1, 23,  23,   1, -729,   1585, -1247,    343;
  1, 76, 230,  76,    1, -14641, 44644, -54230, 30404, -6561;
  ...

Prog

(PARI) T(n, k) = polcoef((1-x)^(n+1)*sum(k=0, n, (2*k+1)^n*x^k), k);
for(n=0, 6, for(k=0, 2*n+1, print1(T(n, k), ", ")));

Crossrefs

Row sums give A000004.

Cf. A060187, A382287.

Sequence in context: A111487 A011505 A373672 * A179975 A019926 A249538

Adjacent sequences: A382286 A382287 A382288 * A382290 A382291 A382292

Keyword

sign,tabf,easy

Author

Seiichi Manyama, Mar 21 2025

Status

approved