A361840 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, 15, 126, 1, 3, 12, 90, 945, 1, 3, 9, 57, 585, 7371, 1, 3, 6, 27, 297, 3969, 58968, 1, 3, 3, 0, 78, 1629, 27657, 480168, 1, 3, 0, -24, -75, 207, 9216, 196290, 3961386, 1, 3, -3, -45, -165, -438, 459, 53217, 1411965, 33011550

Offset

0,3

Links

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

Formula

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

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

Example

Square array begins:
     1,    1,    1,   1,    1,    1, ...
     3,    3,    3,   3,    3,    3, ...
    18,   15,   12,   9,    6,    3, ...
   126,   90,   57,  27,    0,  -24, ...
   945,  585,  297,  78,  -75, -165, ...
  7371, 3969, 1629, 207, -438, -444, ...

Prog

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

Crossrefs

Columns k=0..3 give A004987, A361843, A361844, A361845.

Main diagonal gives A361847.

Cf. A361834, A361839.

Sequence in context: A197272 A306773 A276639 * A361839 A160708 A040173

Adjacent sequences: A361837 A361838 A361839 * A361841 A361842 A361843

Keyword

sign,tabl

Author

Seiichi Manyama, Mar 26 2023

Status

approved