A382287 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} (k+1)^n * x^k.

1, -1, 1, 0, -3, 2, 1, 1, 0, -16, 23, -9, 1, 4, 1, 0, -125, 284, -229, 64, 1, 11, 11, 1, 0, -1296, 4079, -5051, 2869, -625, 1, 26, 66, 26, 1, 0, -16807, 68074, -114546, 98914, -43531, 7776, 1, 57, 302, 302, 57, 1, 0, -262144, 1303567, -2784937, 3243218, -2159662, 776887, -117649

Offset

0,5

Links

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

OEIS Wiki, Eulerian polynomials.

Formula

T(n,k) = A173018(n,k) for 0 <= k <= n.

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

Example

Irregular triangle begins:
  1, -1;
  1,  0, -3,   2;
  1,  1,  0, -16,   23,    -9;
  1,  4,  1,   0, -125,   284,   -229,    64;
  1, 11, 11,   1,    0, -1296,   4079, -5051,    2869,  -625;
  1, 26, 66,  26,    1,     0, -16807, 68074, -114546, 98914, -43531, 7776;
  ...

Prog

(PARI) T(n, k) = polcoef((1-x)^(n+1)*sum(k=0, n, (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. A173018, A382289.

Sequence in context: A155033 A283417 A107889 * A138384 A129172 A355693

Adjacent sequences: A382284 A382285 A382286 * A382288 A382289 A382290

Keyword

sign,tabf,easy

Author

Seiichi Manyama, Mar 20 2025

Status

approved