A228905 G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n*k, k^2) * x^k ).

1, 1, 2, 3, 5, 12, 33, 139, 1251, 10598, 176642, 4720781, 106779821, 5953841083, 373265833332, 23827795512789, 3914313805097976, 548326897932632059, 108647952177920032693, 45931050219457726501030, 14741338951262398648743248, 9489791738688118291360645939

Offset

0,3

Comments

Equals the antidiagonal sums of triangle A228904.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..120

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 12*x^5 + 33*x^6 + 139*x^7 +...
such that, by definition, the logarithm equals (cf. A228832):
log(A(x)) = (1 + x)*x + (1 + 2*x + x^2)*x^2/2 + (1 + 3*x + 15*x^2 + x^3)*x^3/3 + (1 + 4*x + 70*x^2 + 220*x^3 + x^4)*x^4/4 + (1 + 5*x + 210*x^2 + 5005*x^3 + 4845*x^4 + x^5)*x^5/5 +...
More explicitly,
log(A(x)) = x + 3*x^2/2 + 4*x^3/3 + 7*x^4/4 + 31*x^5/5 + 114*x^6/6 + 687*x^7/7 + 8679*x^8/8 + 82948*x^9/9 +...

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, m, binomial(m*k, k^2)*x^k)*x^m/m)+x*O(x^n)), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A228904, A228832.

Cf. variants: A206850, A207137, A206830.

Sequence in context: A233231 A319912 A153987 * A369847 A306038 A008323

Adjacent sequences: A228902 A228903 A228904 * A228906 A228907 A228908

Keyword

nonn

Author

Paul D. Hanna, Sep 07 2013

Status

approved