A303922 Column sums of triangle A303920.

1, 1, 12, 435, 60607, 32465376, 67856416808, 560418604644648, 18418643482653787248, 2416653303692582729686744, 1267452375341631770930186428169, 2658327966985973593187656395635032767, 22300420873364447640210289607043443823426176, 748285604725151189853520504436684719836490370604576

Offset: 0

Author

Paul D. Hanna, May 02 2018

Keywords

• nonn

Comments

G.f. of A303920: (1-y) * Sum_{n>=0} y^n * (1 + x*(1-y)^2)^(n^2) = Sum_{n>=0} Sum_{k=0..2*n} A303920(n,k)*x^n*y^k; the g.f. of this sequence is at y=x, x=1.

Examples

			G.f.: A(x) = 1 + x + 12*x^2 + 435*x^3 + 60607*x^4 + 32465376*x^5 + 67856416808*x^6 + 560418604644648*x^7 + 18418643482653787248*x^8 + ...
such that
A(x)/(1-x) = 1 + x*(2 - 2*x + x^2) + x^2*(2 - 2*x + x^2)^4 + x^3*(2 - 2*x + x^2)^9 + x^4*(2 - 2*x + x^2)^16 + x^5*(2 - 2*x + x^2)^25 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..57

Crossrefs

Cf. A303920, A303921.

Programs

PARI

/* G.f. by Definition: */
{a(n) = my(A = (1-x) * sum(m=0,2*n, x^m * (1 + (1-x)^2  +x*O(x^n) )^(m^2))); polcoeff(A, n,x)}
for(n=0, 20, print1(a(n),", "))

PARI

/* Continued fraction expression: */
{a(n) = my(CF=1, q = 1 + (1-x)^2 +x*O(x^n)); for(k=0, n, CF = 1/(1 - q^(4*n-4*k+1)*x/(1 - q^(2*n-2*k+1)*(q^(2*n-2*k+2) - 1)*x*CF)) ); polcoeff((1-x)*CF, n,x)}
for(n=0, 20, print1(a(n),", "))

PARI

/* G.f. by q-series identity: */
{a(n) = my(A =1, q = 1 + (1-x)^2 +x*O(x^n)); A = (1-x) * sum(m=0,2*n, x^m*q^m * prod(k=1,m, (1 - x*q^(4*k-3)) / (1 - x*q^(4*k-1) +x*O(x^n)) )); polcoeff(A, n,x)}
for(n=0, 20, print1(a(n),", "))

Formula

GENERATING FUNCTIONS.

(1) A(x) = (1-x) * Sum_{n>=0} x^n * (1 + (1-x)^2)^(n^2).

(2) A(x) = (1-x) * Sum_{n>=0} x^n*q^n * Product_{k=1..n} (1 - q^(4*k-3)*x) / (1 - q^(4*k-1)*x) where q = 1 + (1-x)^2, due to a q-series identity.

(3) A(x) = (1-x)/(1 - q*x/(1 - q*(q^2-1)*x/(1 - q^5*x/(1 - q^3*(q^4-1)*x/(1 - q^9*x/(1- q^5*(q^6-1)*x/(1 - q^13*x/(1 - q^7*(q^8-1)*x/(1 - ...))))))))) where q = 1 + (1-x)^2, a continued fraction due to an identity of a partial elliptic theta function.

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Oct 08 2019