A247481 -id:A247481 - cp's OEIS frontend

A247480 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(5n) Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

1, 1, 3, 21, 172, 1557, 14937, 148870, 1523150, 15874211, 167584946, 1784250269, 19082848084, 204183773733, 2174724531143, 22887441573480, 235016048710027, 2294441979279215, 19936497820248076, 118333942636382173, -709004900481995789, -49850788347995316262

Offset: 0

Vaclav Kotesovec, Dec 01 2014

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..260

Cf. A247482 (exponent=0), A247481 (exponent=1), A249934 (exponent=3), A214692 (exponent=4), A214693 (exponent=6), A214694 (exponent=8), A214695 (exponent=10).

Cf. A214690, A214670.

nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1))/AGF^5,{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}]

a(n) ~ c * 12^n * n^(n-2) / (exp(n) * Pi^(2*n)), where c = -sqrt(6) * Pi^3 * exp(5*Pi^2/24)/24 = -24.7341070998048267... - Vaclav Kotesovec, Dec 01 2014, updated Aug 22 2017

A247482 G.f. A(x) satisfies: x = Sum_{n>=1} Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

Original entry on oeis.org

1, 1, -2, 1, -3, -18, -124, -1174, -12150, -141536, -1816780, -25461723, -386593670, -6320496592, -110711177281, -2068814967831, -41089562943757, -864563028340432, -19214971769126974, -449887669808788433, -11069673481210168218, -285604488897863640237

Offset: 0

Vaclav Kotesovec, Dec 01 2014

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..370

Cf. A247481 (exponent=1), A249934 (exponent=3), A214692 (exponent=4), A247480 (exponent=5), A214693 (exponent=6), A214694 (exponent=8), A214695 (exponent=10).

Cf. A214690, A214670.

nmax = 20; aa = ConstantArray[0,nmax]; aa[[1]] = 1; Do[AGF = 1+Sum[aa[[n]]*x^n,{n,1,j-1}]+koef*x^j; sol=Solve[SeriesCoefficient[Sum[Product[(1-1/AGF^(2m-1)),{m,1,k}],{k,1,j}],{x,0,j}]==0,koef][[1]]; aa[[j]]=koef/.sol[[1]],{j,2,nmax}]; Flatten[{1,aa}]

{a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0);
A[#A]=-polcoeff(sum(m=1, #A, prod(k=1, m, 1-1/Ser(A)^(2*k-1))), #A-1)); A[n+1]}
for(n=0, 25, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 17 2024, after Paul D. Hanna

a(n) ~ c * 12^n * n^(n+1/2) / (exp(n) * Pi^(2*n)), where c = -12 / (Pi^(3/2) * exp(5*Pi^2/24)) = -0.275723765924812729... - Vaclav Kotesovec, Dec 01 2014, updated Aug 22 2017

cp's OEIS Frontend

A247480 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(5n) Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

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A247482 G.f. A(x) satisfies: x = Sum_{n>=1} Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

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cp's OEIS Frontend

A247480 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(5*n) * Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

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A247482 G.f. A(x) satisfies: x = Sum_{n>=1} Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).

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Mathematica

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A247480 G.f. A(x) satisfies: x = Sum_{n>=1} 1/A(x)^(5n) Product_{k=1..n} (1 - 1/A(x)^(2*k-1)).