A145267 -id:A145267 - cp's OEIS frontend

A192477 G.f. satisfies: A(x) = xProduct_{n>=1} (1 + xA(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

Paul D. Hanna, Jul 01 2011

nonn

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).

			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)) +...

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

Cf. A145267, A192478.

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 *)

{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),", "))

{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)}

{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)}

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

A302287 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x))^k.

Original entry on oeis.org

1, 1, 3, 10, 31, 102, 342, 1167, 4046, 14213, 50464, 180847, 653296, 2376406, 8697194, 32002219, 118322499, 439364380, 1637827543, 6126870808, 22993190147, 86542625565, 326607659370, 1235650643059, 4685502714403, 17804713119018, 67790202024365, 258579199501709, 988012193672223

Offset: 0

Ilya Gutkovskiy, Apr 04 2018

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 10*x^3 + 31*x^4 + 102*x^5 + 342*x^6 + 1167*x^7 + 4046*x^8 + 14213*x^9 + 50464*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x))^2 * (1 + x^3*A(x))^3 * (1 + x^4*A(x))^4 * ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400

Cf. A026007, A145267, A298260, A301456, A302171.

nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x])^k, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)

a(n) ~ c * d^n / n^(3/2), where d = 4.01604513838270620496843653760987690323... and c = 2.07544072297996637757124624302382219... - Vaclav Kotesovec, Sep 27 2023

Radius of convergence r = 0.2490011853807768883971843288180859269 = 1/d and A(r) = 3.261386924996517219078267128734843819... satisfy (1) A(r) = 1 / Sum_{n>=1} n*r^n/(1 + r^n*A(r)) and (2) A(r) = Product_{n>=1} (1 + r^n*A(r))^n. - Paul D. Hanna, Mar 02 2024

A302289 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + kx^kA(x)).

Original entry on oeis.org

1, 1, 3, 10, 30, 98, 323, 1083, 3684, 12710, 44272, 155608, 551259, 1965952, 7052990, 25436711, 92168542, 335376653, 1224991077, 4489818110, 16507728007, 60868469848, 225030777305, 833961333273, 3097594423724, 11529400593846, 42996077073284, 160632616725238, 601132116489719, 2253153800577748

Offset: 0

Ilya Gutkovskiy, Apr 04 2018

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 10*x^3 + 30*x^4 + 98*x^5 + 323*x^6 + 1083*x^7 + 3684*x^8 + 12710*x^9 + 44272*x^10 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + 2*x^2*A(x)) * (1 + 3*x^3*A(x)) * (1 + 4*x^4*A(x)) * ...

Cf. A022629, A145267, A301456, A301577, A301578, A302287, A302288.

A192785 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n)(1 + x^(2n)*A(x)^2).

Original entry on oeis.org

1, 1, 2, 5, 11, 29, 75, 203, 557, 1561, 4427, 12706, 36819, 107576, 316579, 937471, 2791487, 8352973, 25104573, 75749240, 229379444, 696851166, 2123304184, 6487295518, 19870096689, 61001089214, 187673207413, 578532522637, 1786712575547

Offset: 0

Paul D. Hanna, Jul 10 2011

nonn

Related q-series identity of Lebesgue:

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

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 29*x^5 + 75*x^6 + ...
The g.f. A = A(x) satisfies the following relations:
A = (1+x)*(1+x^2*A^2)* (1+x^2)*(1+x^4*A^2)* (1+x^3)*(1+x^6*A^2)* ...
A = 1 + x*(1+x*A^2)/(1-x) + x^3*(1+x*A^2)*(1+x^2*A^2)/((1-x)*(1-x^2)) + x^6*(1+x*A^2)*(1+x^2*A^2)*(1+x^3*A^2)/((1-x)*(1-x^2)*(1-x^3)) + ...

Cf. A190822, A145267.

nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k)*(1 + x^(2*k)*A[x]^2), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Mar 04 2024 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, Sqrt[(-r*s*(1 + s^2) * Log[r]^2 * ((s + s^3)*Derivative[0, 1][QPochhammer][-1, r] + r*QPochhammer[-1, r]^2 * Derivative[0, 1][QPochhammer][-s^2, r^2]))/ (2*Pi * QPochhammer[-1, r]*(4*s^2*Log[r]^2 + (1 + s^2)^2 * QPolyGamma[1, Log[-s^2]/(2*Log[r]), r^2]))]} /. FindRoot[{(QPochhammer[-1, r]*QPochhammer[-s^2, r^2])/(2 + 2*s^2) == s, 1 + 3*s^2 + (1 + s^2)*((Log[1 - r^2] + QPolyGamma[0, Log[-s^2]/(2*Log[r]), r^2])/Log[r]) == 0}, {r, 1/3}, {s, 2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Mar 04 2024 *)

{a(n) = my(A=1+x); for(i=1, n, A = prod(m=1, n, (1 + x^m)*(1 + x^(2*m)*A^2 +x*O(x^n)))); polcoeff(A, n)}

{a(n) = my(A=1+x); for(i=1, n, A = 1 + sum(m=1, sqrtint(2*n), x^(m*(m+1)/2)*prod(k=1, m, (1 + A^2*x^k)/(1 - x^k+x*O(x^n))))); polcoeff(A, n)}

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*Product_{k=1..n} (1 + x^k*A(x)^2)/(1 - x^k) due to a Lebesgue identity.
From Vaclav Kotesovec, Mar 04 2024: (Start)
Let A(x) = y, then 2*y*(1 + y^2) = QPochhammer(-1, x) * QPochhammer(-y^2, x^2).
a(n) ~ c * d^n / n^(3/2), where
d = 3.25215123067662173854186425074452291189580485719079882122325713176...,
c = 1.30862302149248708183423553797270804891358016970005788341511105232...
Radius of convergence:
r = 1/d = 0.307488775604062671485504670197489134974315527740973676344144395...
A(r) = s = 2.80682163771231540175973628784430270489737819467327067575665055...
(End)
The values r and A(r) given above also satisfy A(r) = 1 / sqrt( Sum_{n>=1} 2*r^(2*n)/(1 + r^(2*n)*A(r)^2) ). - Paul D. Hanna, Mar 06 2024

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

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A302287 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x))^k.

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A302289 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + k*x^k*A(x)).

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A192785 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n)*(1 + x^(2*n)*A(x)^2).

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

A302289 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + kx^kA(x)).

A192785 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n)(1 + x^(2n)*A(x)^2).