A394823 - OEIS

%I #10 Apr 03 2026 10:10:41

%S 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,

%T 275,6,0,1,2730,39039,45760,10010,546,7,0,1,10922,355446,784212,

%U 307450,31668,980,8,0,1,43690,3215397,13021320,8666515,1508988,84966,1632,9

%N 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)).

%e Triangle starts:

%e   [0]  1;

%e   [1]  0, 1;

%e   [2]  0, 1,     2;

%e   [3]  0, 1,    10,       3;

%e   [4]  0, 1,    42,      42,        4;

%e   [5]  0, 1,   170,     441,      120,       5;

%e   [6]  0, 1,   682,    4224,     2508,     275,       6;

%e   [7]  0, 1,  2730,   39039,    45760,   10010,     546,     7;

%e   [8]  0, 1, 10922,  355446,   784212,  307450,   31668,   980,    8;

%e   [9]  0, 1, 43690, 3215397, 13021320, 8666515, 1508988, 84966, 1632, 9;

%p u := sqrt(2*x*(cosh(y^(1/2)) - 1)): ser := series(1 + u*sinh(u)/2, y, 11):

%p ycoeff := n -> expand((2*n)!*simplify(coeff(ser, y, n))):

%p row := n -> local k; seq(coeff(ycoeff(n), x, k), k = 0..n):

%p seq(print(row(n)), n = 0..9);

%o (Python)

%o from functools import cache

%o @cache

%o def cfact2(n, j):

%o     if j == 0: return int(n == 0)

%o     if j < 1 or j > n: return 0

%o     return cfact2(n-1, j-1) + j * j * cfact2(n-1, j)

%o def T(n, j): return 1 if n == 0 else j*cfact2(n, j)

%o for n in range(10): print([T(n, j) for j in range(n + 1)])

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

%K nonn,tabl

%O 0,6

%A _Peter Luschny_, Apr 03 2026