A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 + x)^k)^(1/3).

1, 1, 3, 1, 3, 18, 1, 3, 21, 126, 1, 3, 24, 162, 945, 1, 3, 27, 201, 1341, 7371, 1, 3, 30, 243, 1809, 11529, 58968, 1, 3, 33, 288, 2352, 16893, 101619, 480168, 1, 3, 36, 336, 2973, 23607, 161676, 911466, 3961386, 1, 3, 39, 387, 3675, 31818, 242757, 1574289, 8281737, 33011550

Offset: 0

Author

Seiichi Manyama, Mar 26 2023

Keywords

• nonn
• tabl

Examples

			Square array begins:
     1,     1,     1,     1,     1,     1, ...
     3,     3,     3,     3,     3,     3, ...
    18,    21,    24,    27,    30,    33, ...
   126,   162,   201,   243,   288,   336, ...
   945,  1341,  1809,  2352,  2973,  3675, ...
  7371, 11529, 16893, 23607, 31818, 41676, ...
		

Crossrefs

Columns k=0..3 give A004987, A180400, A361841, A361842.

Main diagonal gives A361846.

Cf. A361830, A361840.

Programs

PARI

T(n,k) = sum(j=0, n, (-9)^j*binomial(-1/3, j)*binomial(k*j, n-j));

Formula

n*T(n,k) = 3 * Sum_{j=0..k} binomial(k,j)*(3*n-2-2*j)*T(n-1-j,k) for n > k.

T(n,k) = Sum_{j=0..n} (-9)^j * binomial(-1/3,j) * binomial(k*j,n-j).