A177486 G.f. satisfies: A(x) = Sum_{n>=0} x^(n(n+1)/2) * (1 + x*A(x))^n.

1, 1, 1, 2, 3, 5, 8, 15, 26, 46, 81, 149, 272, 499, 916, 1701, 3168, 5921, 11095, 20880, 39407, 74558, 141386, 268776, 512059, 977419, 1869033, 3580161, 6868969, 13198565, 25396248, 48931453, 94395624, 182317031, 352522329, 682346776

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Formula

Let A = g.f. A(x), then A satisfies:

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

(2) A = 1/(1- x*(1+xA)/(1- (x^2-x)*(1+xA)/(1- x^3*(1+xA)/(1- (x^4-x^2)*(1+xA)/(1- x^5*(1+xA)/(1- (x^6-x^3)*(1+xA)/(1- ...))))))) due to a continued fraction of a partial theta function.

a(n) ~ c * d^n / n^(3/2), where d = 2.01316244..., c = 3.801615... . - Vaclav Kotesovec, Aug 25 2014

Example

G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 8*x^6 + 15*x^7 +...
Let A = A(x) then A satisfies:
A = 1 + (x + x^2*A) + (x^3 + 2*x^4*A + x^5*A^2) + (x^6 + 3*x^7*A + 3*x^8*A^2 + x^9*A^3) + (x^10 + 4*x^11*A + 6*x^12*A^2 + 4*x^13*A^3 + x^14*A^4) +...
in which all the binomial coefficients appear in succession.

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, (1+x*A+x*O(x^n))^m*x^(m*(m+1)/2))); polcoeff(A, n)}
(PARI) /* q-series expansion: */ {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m*(1+x*A)^m*prod(k=1, m, (1-(1+x*A)*x^(2*k-1))/(1-(1+x*A)*x^(2*k)+x*O(x^n))))); polcoeff(A, n)}

Crossrefs

Cf. A177487.

Sequence in context: A151518 A362785 A082095 * A006982 A054539 A026702

Adjacent sequences: A177483 A177484 A177485 * A177487 A177488 A177489

Keyword

nonn

Author

Paul D. Hanna, May 09 2010

Status

approved