A192477 G.f. satisfies: A(x) = x*Product_{n>=1} (1 + x*A(x)^n).

1, 0, 1, 1, 2, 5, 9, 23, 49, 120, 279, 682, 1654, 4079, 10129, 25277, 63639, 160685, 408373, 1041197, 2666364, 6850405, 17657214, 45644461, 118303445, 307385607, 800463683, 2088900834, 5461793800, 14306839474, 37539357792, 98655089606

Offset

1,5

Comments

Related q-series identity (Euler):

Product_{n>=1} (1+x*q^n) = Sum_{n>=0} x^n*q^(n*(n+1)/2) / Product_{k=1..n} (1-q^k); here q=A(x).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..300

Formula

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

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

a(n) ~ c * d^n / n^(3/2), where d = 2.7572424362046888622202089939389819515998799032935772914495266456251... and c = 0.141814541288727417106640836565322805487015901140336362320896774237... - Vaclav Kotesovec, Sep 29 2023

Example

G.f.: A(x) = x + x^3 + x^4 + 2*x^5 + 5*x^6 + 9*x^7 + 23*x^8 + 49*x^9 +...
The g.f. A = A(x) satisfies the relations:
A = x*(1 + x*A)*(1 + x*A^2)*(1 + x*A^3)*(1 + x*A^4)*...
A = x*(1 + x*A/(1-A) + x^2*A^3/((1-A)*(1-A^2)) + x^3*A^6/((1-A)*(1-A^2)*(1-A^3)) + x^4*A^10/((1-A)*(1-A^2)*(1-A^3)*(1-A^4)) +...)
A = x*(1+x*A) + x^2*A^2*(1+x*A^3)*(1+x*A)/(1-A) + x^3*A^7*(1+x*A^5)*(1+x*A)*(1+x*A^2)/((1-A)*(1-A^2)) + x^4*A^15*(1+x*A^7)*(1+x*A)*(1+x*A^2)*(1+x*A^3)/((1-A)*(1-A^2)*(1-A^3)) +...

Mathematica

nmax = 40; A[_] = 0; Do[A[x_] = x*Product[1 + x*A[x]^k, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; Rest[CoefficientList[A[x], x]] (* Vaclav Kotesovec, Sep 29 2023 *)
(* Calculation of constants {d, c}: *) {1/r, Sqrt[-s*((1 + r)*Log[1 - s] - Log[s] + (1 + r)*QPolyGamma[0, Log[-r]/Log[s], s]) / (2*Pi*r*Log[s] * Derivative[0, 2][QPochhammer][-r, s])]} /. FindRoot[{r*QPochhammer[-r, s] == s*(1 + r), r*Derivative[0, 1][QPochhammer][-r, s] == 1 + r}, {r, 1/3}, {s, 1/2}, WorkingPrecision -> 120] // Chop (* Vaclav Kotesovec, Sep 29 2023 *)

Prog

(PARI) {a(n) = my(A=x+x^2); for(i=1, n, A = x*prod(m=1, n, 1+x*A^m +x*O(x^n))); polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))
(PARI) {a(n) = my(A=x+x^2); for(i=1, n, A = x*sum(m=0, n, x^m*A^(m*(m+1)/2)/prod(k=1, m, 1-A^k +x*O(x^n)))); polcoeff(A, n)}
(PARI) {a(n) = my(A=x+x^2); for(i=1, n, A = x*sum(m=0, n, x^m*A^(m*(3*m+1)/2)*(1+x*A^(2*m+1))*prod(k=1, m, (1+x*A^k)/(1-A^k +x*O(x^n))))); polcoeff(A, n)}

Crossrefs

Cf. A145267, A192478.

Sequence in context: A374244 A088356 A246350 * A288109 A047044 A109621

Adjacent sequences: A192474 A192475 A192476 * A192478 A192479 A192480

Keyword

nonn

Author

Paul D. Hanna, Jul 01 2011

Status

approved