A299435 G.f.: Sum_{n>=0} binomial((n+1)^2, n)/(n+1) * x^n / (1 + x)^((n+1)^2).

1, 1, 5, 51, 791, 16711, 449575, 14738537, 570860449, 25534320961, 1296145448621, 73644069770107, 4631766294581959, 319523289664700279, 23992478864877747151, 1948216141720780468561, 170121586262631029818433, 15897659114382366967974145, 1583109774987253349677203349, 167363833662976153803805436291, 18721216520653602533835176495671

Offset

0,3

Comments

Compare to: 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m.

Links

Table of n, a(n) for n=0..20.

Example

G.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 791*x^4 + 16711*x^5 + 449575*x^6 + 14738537*x^7 + 570860449*x^8 + 25534320961*x^9 + 1296145448621*x^10 + ...
such that
A(x) = 1/(1+x) + C(4,1)/2*x/(1+x)^4 + C(9,2)/3*x^2/(1+x)^9 + C(16,3)/4*x^3/(1+x)^16 + C(25,4)/5*x^4/(1+x)^25 + C(36,5)/6*x^5/(1+x)^36 + C(49,6)/7*x^6/(1+x)^49 + ...
more explicitly,
A(x) = 1/(1+x) + 2*x/(1+x)^4 + 12*x^2/(1+x)^9 + 140*x^3/(1+x)^16 + 2530*x^4/(1+x)^25 + 62832*x^5/(1+x)^36 + 1997688*x^6/(1+x)^49 + ... + A091144(n+1)*x^n/(1+x)^((n+1)^2) + ...

Prog

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

Crossrefs

Cf. A298695, A298696, A091144.

Sequence in context: A343674 A318192 A343687 * A095839 A234290 A107669

Adjacent sequences: A299432 A299433 A299434 * A299436 A299437 A299438

Keyword

nonn

Author

Paul D. Hanna, Feb 09 2018

Status

approved