A145267 G.f. satisfies A(x) = Product_{k>0} (1+x^k*A(x)).

1, 1, 2, 5, 12, 30, 77, 201, 532, 1427, 3868, 10579, 29161, 80931, 225954, 634197, 1788453, 5064877, 14398536, 41074364, 117541744, 337337862, 970704394, 2800059428, 8095161902, 23452565124, 68076579332, 197965830430

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Formula

G.f. satisfies: A(x) = Sum_{n>=0} x^(n*(3n+1)/2)*A(x)^n*(1 + x^(2n+1)*A(x))*Product_{k=1..n} (1 + x^k*A(x))/(1-x^k) due to Sylvester's identity. - Paul D. Hanna, May 20 2011

G.f. satisfies: A(x) = Sum_{n>=0} x^(n*(n+1)/2)*A(x)^n / Product_{k=1..n} (1-x^k). - Paul D. Hanna, Jul 01 2011

a(n) ~ c * d^n / n^(3/2), where d = 3.060735101304296413235... and c = 2.45762465379034328... - Vaclav Kotesovec, Aug 12 2021

Radius of convergence r = 0.32671889820646736561... = 1/d and A(r) = 3.6673575238633912689... satisfy (1) A(r) = 1 / Sum_{n>=1} r^n/(1 + r^n*A(r)) and (2) A(r) = Product_{n>=1} (1 + r^n*A(r)). - Paul D. Hanna, Mar 02 2024

Example

From Paul D. Hanna, May 20 2011: (Start)
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 30*x^5 + 77*x^6 +...
G.f.: A(x) = (1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x))*(1+x^4*A(x))*...
G.f.: A(x) = (1+x*A(x)) + x^2*A(x)*(1 + x^3*A(x))*(1+x*A(x))/(1-x) + x^7*A(x)^2*(1 + x^5*A(x))*(1+x*A(x))*(1+x^2*A(x))/((1-x)*(1-x^2)) + x^15*A(x)^3*(1 + x^7*A(x))*(1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x))/((1-x)*(1-x^2)*(1-x^3)) +... (End)
G.f.: A(x) = 1 + x*A(x)/(1-x) + x^3*A(x)^2/((1-x)*(1-x^2)) + x^6*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^10*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +... - Paul D. Hanna, Jul 01 2011

Mathematica

nmax = 30; A[_] = 0; Do[A[x_] = Product[1 + x^k*A[x], {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{QPochhammer[-s, r] == s*(1 + s), Log[1 - r] + ((1 + 2*s)*Log[r])/(1 + s) + QPolyGamma[0, Log[-s]/Log[r], r] == 0}, {r, 1/3}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 26 2023 *)

Prog

(PARI) {a(n)=local(A=1+x); for(i=1, n, A=prod(m=1, n, (1+A*x^m+x*O(x^n)))); polcoeff(A, n)}  /* Paul D. Hanna, May 20 2011 */
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^(m*(3*m+1)/2)*A^m*(1 + x^(2*m+1)*A)*prod(k=1, m, (1+A*x^k)/(1-x^k+x*O(x^n))))); polcoeff(A, n)}  /* Paul D. Hanna, May 20 2011 */
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^(m*(m+1)/2)*A^m/prod(k=1, m, 1-x^k +x*O(x^n)))); polcoeff(A, n)} /* Paul D. Hanna, Jul 01 2011 */

Crossrefs

Cf. A145268, A190822.

Sequence in context: A253831 A024851 A188378 * A103287 A136704 A120895

Adjacent sequences: A145264 A145265 A145266 * A145268 A145269 A145270

Keyword

nonn

Author

Vladeta Jovovic, Oct 05 2008

Extensions

More terms from Max Alekseyev, Jan 31 2010

Status

approved