A376878 Triangle read by rows: T(n, k) = n^k * n! * [x^k][y^n]((sec(y) + tan(y)) * exp(x*y)).

1, 1, 1, 1, 4, 4, 2, 9, 27, 27, 5, 32, 96, 256, 256, 16, 125, 500, 1250, 3125, 3125, 61, 576, 2700, 8640, 19440, 46656, 46656, 272, 2989, 16464, 60025, 168070, 352947, 823543, 823543, 1385, 17408, 109312, 458752, 1433600, 3670016, 7340032, 16777216, 16777216

Offset: 0

Author

Peter Luschny, Oct 13 2024

Keywords

• nonn
• tabl

Examples

			Triangle starts:
  [0]    1;
  [1]    1,     1;
  [2]    1,     4,      4;
  [3]    2,     9,     27,     27;
  [4]    5,    32,     96,    256,     256;
  [5]   16,   125,    500,   1250,    3125,    3125;
  [6]   61,   576,   2700,   8640,   19440,   46656,   46656;
  [7]  272,  2989,  16464,  60025,  168070,  352947,  823543,   823543;
  [8] 1385, 17408, 109312, 458752, 1433600, 3670016, 7340032, 16777216, 16777216;
		

Crossrefs

Cf. A000111, A000312, A079901, A109449, A292976 (row sums).

Programs

Maple

P := n -> coeff(series((sec(y) + tan(y)) * exp(x*y), y, 12), y, n):
seq(seq(coeff(P(n), x,  k) * n^k * n!, k = 0..n), n = 0..8);
T := (n, k) -> ifelse(n = k, n^n, (-1)^binomial(n - k, 2)*n^k*binomial(n, k)*(euler(n - k) - euler(n - k, 0)*2^(n - k))):
seq(print([n], seq(T(n, k), k = 0..n)), n = 0..8);

Python

from math import comb, isqrt
from sympy import bernoulli, euler
def A000111(n): return abs(((1<A376878(n): return comb(a:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)),b:=n-comb(a+1,2))*a**b*A000111(a-b) # Chai Wah Wu, Nov 13 2024

Formula

T(n, k) = (-1)^binomial(n-k, 2)*n^k*binomial(n, k)*(Euler(n-k) - Euler(n-k, 0)*2^(n - k)) for 0 <= k < n and n^n for n = k.

T(n, k) = n^k*A109449(n, k) = n^k*binomial(n, k)*A000111(n - k).