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A192620 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))^2/(1 - x^n*A(x))^2. 4
1, 4, 28, 224, 1948, 17928, 171776, 1695872, 17133436, 176297668, 1841222776, 19467629120, 207978652416, 2241618514120, 24345336854400, 266168049520832, 2927074607294300, 32356419163487336, 359330087240388828, 4007079691584624576 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Related q-series (Heine) identity:

1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n)); here q=x, x=x*A(x), y=z=1.

LINKS

Table of n, a(n) for n=0..19.

FORMULA

G.f. A(x) satisfies:

(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^n * Product_{k=1..n} (1+x^(k-1))^2/((1-x^k*A(x))*(1-x^k), due to the Heine identity.

(2) A(x)^(1/2) = 1 + Sum_{n>=1} x^n*A(x)^n * Product_{k=1..n} (1+x^(k-1))/(1-x^k), due to the q-binomial theorem.

Equals the self-convolution of A192621.

EXAMPLE

G.f.: A(x) = 1 + 4*x + 28*x^2 + 224*x^3 + 1948*x^4 + 17928*x^5 +...

The g.f. A = A(x) satisfies the following relations:

(0) A = (1+x*A)^2/(1-x*A)^2 * (1+x^2*A)^2/(1-x^2*A)^2 * (1+x^3*A)^2/(1-x^3*A)^2 *...

(1) A = 1 + 4*x*A/((1-x*A)*(1-x)) + 4*x^2*A^2*(1+x)^2/((1-x*A)*(1-x^2*A)*(1-x)*(1-x^2)) + 4*x^3*A^3*(1+x)^2*(1+x^2)^2/((1-x*A)*(1-x^2*A)*(1-x^3*A)*(1-x)*(1-x^2)*(1-x^3)) +...

(2) A^(1/2) = 1 + 2*x*A/(1-x) + 2*x^2*A^2*(1+x)/((1-x)*(1-x^2)) + 2*x^3*A^3*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3)) +...

PROG

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(k=1, n, (1+x^k*A)^2/(1-x^k*A+x*O(x^n))^2)); polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A^m*prod(k=1, m, (1+x^(k-1))^2/((1-x^k*A+x*O(x^n))*(1-x^k))))); polcoeff(A, n)}

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=(1+sum(m=1, n, x^m*A^m*prod(k=1, m, (1+x^(k-1))/(1-x^k+x*O(x^n)))))^2); polcoeff(A, n)}

CROSSREFS

Cf. A192621, A192622, A192623, A192624, A192625.

Sequence in context: A026033 A005810 A121203 * A180708 A191094 A220876

Adjacent sequences:  A192617 A192618 A192619 * A192621 A192622 A192623

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 06 2011

STATUS

approved

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Last modified May 6 23:37 EDT 2021. Contains 343600 sequences. (Running on oeis4.)