A108290 Triangle, read by rows, such that row n equals the inverse binomial transform of row n of table A060543, where A060543(n,k) = C(n+n*k+k, n*k+k).

1, 1, 2, 1, 9, 9, 1, 34, 96, 64, 1, 125, 750, 1250, 625, 1, 461, 5265, 16470, 19440, 7776, 1, 1715, 35329, 184877, 386561, 352947, 117649, 1, 6434, 232288, 1913408, 6307840, 9863168, 7340032, 2097152, 1, 24309, 1513656, 18921924, 92365758, 220016574

Offset

0,3

Comments

Row sums form A108292. Main diagonal is A000169(n) = (n+1)^n. Triangle with rows reversed is A108291.

The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (1+n*x)*(2+n*x)*...*(n-1+n*x)/(n-1)! in the basis made of the binomial(x+i,i). - F. Chapoton, Nov 01 2022

Links

Table of n, a(n) for n=0..41.

Example

BINOMIAL[1, 9, 9] = {1, 10, 28, 55, 91, 136, 190, 253, ...}.
BINOMIAL[1, 34, 96, 64] = {1, 35, 165, 455, 969, 1771, 2925, ...}.
BINOMIAL[1, 125, 750, 1250, 625] = {1, 126, 1001, 3876, 10626, ...}.
Triangle begins:
  1;
  1,    2;
  1,    9,      9;
  1,   34,     96,      64;
  1,  125,    750,    1250,     625;
  1,  461,   5265,   16470,   19440,    7776;
  1, 1715,  35329,  184877,  386561,  352947,  117649;
  1, 6434, 232288, 1913408, 6307840, 9863168, 7340032, 2097152; ...

Prog

(PARI) {T(n, k)=local(X=x+x*O(x^k)); polcoeff(sum(j=0, n, binomial(n+n*j+j, n*j+j)*(x/(1+X))^j)/(1+X), k)}

Crossrefs

Cf. A108267, A060543, A108290, A000169.

Sequence in context: A243261 A230358 A228721 * A308804 A364709 A108291

Adjacent sequences: A108287 A108288 A108289 * A108291 A108292 A108293

Keyword

nonn,tabl

Author

Paul D. Hanna, May 31 2005

Status

approved