A177487 G.f. satisfies: A(x) = Sum_{n>=0} x^(n(n+1)/2) * (A(x) + x)^n.

1, 1, 2, 3, 7, 15, 30, 66, 154, 352, 800, 1864, 4415, 10474, 24935, 59845, 144527, 350271, 852002, 2080955, 5100468, 12537191, 30900833, 76363218, 189162367, 469590870, 1168098275, 2911137073, 7267923924, 18174761288, 45519170649

Offset

0,3

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

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

a(n) ~ c * d^n / n^(3/2), where d = 2.6256644..., c = 2.210416... . - Vaclav Kotesovec, Aug 25 2014

Example

G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 7*x^4 + 15*x^5 + 30*x^6 +...
Let A = A(x) then A satisfies:
A = 1 + (x*A + x^2) + (x^3*A^2 + 2*x^4*A + x^5) + (x^6*A^3 + 3*x^7*A^2 + 3*x^8*A + x^9) + (x^10*A^4 + 4*x^11*A^3 + 6*x^12*A^2 + 4*x^13*A + x^14) +...
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, (A+x+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*(A+x)^m*prod(k=1, m, (1-(A+x)*x^(2*k-1))/(1-(A+x)*x^(2*k)+x*O(x^n))))); polcoeff(A, n)}

Crossrefs

Cf. A177486.

Sequence in context: A020873 A049958 A298403 * A153010 A076993 A076698

Adjacent sequences: A177484 A177485 A177486 * A177488 A177489 A177490

Keyword

nonn

Author

Paul D. Hanna, May 09 2010

Status

approved