A192477 - OEIS

%I #17 Sep 29 2023 10:54:24

%S 1,0,1,1,2,5,9,23,49,120,279,682,1654,4079,10129,25277,63639,160685,

%T 408373,1041197,2666364,6850405,17657214,45644461,118303445,307385607,

%U 800463683,2088900834,5461793800,14306839474,37539357792,98655089606

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

%C Related q-series identity (Euler):

%C 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).

%H Vaclav Kotesovec, <a href="/A192477/b192477.txt">Table of n, a(n) for n = 1..300</a>

%F 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).

%F 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.

%F a(n) ~ c * d^n / n^(3/2), where d = 2.7572424362046888622202089939389819515998799032935772914495266456251... and c = 0.141814541288727417106640836565322805487015901140336362320896774237... - _Vaclav Kotesovec_, Sep 29 2023

%e G.f.: A(x) = x + x^3 + x^4 + 2*x^5 + 5*x^6 + 9*x^7 + 23*x^8 + 49*x^9 +...

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

%e A = x*(1 + x*A)*(1 + x*A^2)*(1 + x*A^3)*(1 + x*A^4)*...

%e 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)) +...)

%e 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)) +...

%t 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 *)

%t (* 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 *)

%o (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)}

%o for(n=0,35,print1(a(n),", "))

%o (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)}

%o (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)}

%Y Cf. A145267, A192478.

%K nonn

%O 1,5

%A _Paul D. Hanna_, Jul 01 2011