A394823 Triangle read by rows: T(n, k) = (2*n)! * [x^k] [y^n] (1 + u*sinh(u)/2) where u = sqrt(2*x*(cosh(y^(1/2)) - 1)).

1, 0, 1, 0, 1, 2, 0, 1, 10, 3, 0, 1, 42, 42, 4, 0, 1, 170, 441, 120, 5, 0, 1, 682, 4224, 2508, 275, 6, 0, 1, 2730, 39039, 45760, 10010, 546, 7, 0, 1, 10922, 355446, 784212, 307450, 31668, 980, 8, 0, 1, 43690, 3215397, 13021320, 8666515, 1508988, 84966, 1632, 9

Offset

0,6

Links

Table of n, a(n) for n=0..54.

Example

Triangle starts:
  [0]  1;
  [1]  0, 1;
  [2]  0, 1,     2;
  [3]  0, 1,    10,       3;
  [4]  0, 1,    42,      42,        4;
  [5]  0, 1,   170,     441,      120,       5;
  [6]  0, 1,   682,    4224,     2508,     275,       6;
  [7]  0, 1,  2730,   39039,    45760,   10010,     546,     7;
  [8]  0, 1, 10922,  355446,   784212,  307450,   31668,   980,    8;
  [9]  0, 1, 43690, 3215397, 13021320, 8666515, 1508988, 84966, 1632, 9;

Maple

u := sqrt(2*x*(cosh(y^(1/2)) - 1)): ser := series(1 + u*sinh(u)/2, y, 11):
ycoeff := n -> expand((2*n)!*simplify(coeff(ser, y, n))):
row := n -> local k; seq(coeff(ycoeff(n), x, k), k = 0..n):
seq(print(row(n)), n = 0..9);

Prog

(Python)
from functools import cache
@cache
def cfact2(n, j):
    if j == 0: return int(n == 0)
    if j < 1 or j > n: return 0
    return cfact2(n-1, j-1) + j * j * cfact2(n-1, j)
def T(n, j): return 1 if n == 0 else j*cfact2(n, j)
for n in range(10): print([T(n, j) for j in range(n + 1)])

Crossrefs

Cf. A108678 (subdiagonal), A020988 (column 1), A394822 (row sums), A394813 (row reversed and (1,1)-based).

Sequence in context: A219034 A372244 A256116 * A185410 A264676 A091803

Adjacent sequences: A394820 A394821 A394822 * A394824 A394825 A394826

Keyword

nonn,tabl

Author

Peter Luschny, Apr 03 2026

Status

approved