A192623 -id:A192623 - cp's OEIS frontend

A192622 G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)A(x)^n)^2/(1 - x^(n+1)A(x)^n)^2.

1, 4, 12, 48, 220, 1080, 5600, 30112, 166300, 937620, 5374200, 31221488, 183430656, 1087975256, 6505878592, 39179738400, 237412139260, 1446488046824, 8855937880108, 54455375407504, 336159421649528, 2082508824181856, 12942736191473792

Offset: 0

Paul D. Hanna, Jul 06 2011

nonn

Related q-series (Heine) identity:

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

			G.f.: A(x) = 1 + 4*x + 12*x^2 + 48*x^3 + 220*x^4 + 1080*x^5 +...
The g.f. A = A(x) satisfies the following relations:
A = (1+x)^2/(1-x)^2 * (1+x^2*A)^2/(1-x^2*A)^2 * (1+x^3*A^2)^2/(1-x^3*A^2)^2 *...
A = 1 + 4*x/((1-x)*(1-x*A)) + 4*x^2*(1+x*A)^2/((1-x)*(1-x*A)*(1-x^2*A)*(1-x^2*A^2)) + 4*x^3*(1+x*A)^2*(1+x^2*A^2)^2/((1-x)*(1-x*A)*(1-x^2*A)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x^3*A^3)) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..160

Cf. A192623, A192620, A192624.

(* Calculation of constants {d,c}: *) Chop[{1/r, s*Sqrt[(QPochhammer[r, r*s]*(Log[r*s] - 2*QPolyGamma[0, Log[-r]/Log[r*s], r*s] + 2*QPolyGamma[0, Log[r]/Log[r*s], r*s]))/(Pi* Log[r*s]*(QPochhammer[r, r*s] - 4*r*s*(Derivative[0, 1][QPochhammer][r, r*s] - r*Sqrt[s]*Derivative[0, 2][QPochhammer][-r, r*s] + r*s*Derivative[0, 2][QPochhammer][r, r*s])))]} /. FindRoot[{QPochhammer[-r, r*s]^2/QPochhammer[r, r*s]^2 == s, QPochhammer[r, r*s]^2 + 2*r*s*QPochhammer[r, r*s] * Derivative[0, 1][QPochhammer][r, r*s] == 2*r*QPochhammer[-r, r*s] * Derivative[0, 1][QPochhammer][-r, r*s]}, {r, 1/6}, {s, 3}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Jun 30 2025 *)

{a(n)=local(A=1+x);for(i=1,n,A=prod(k=0,n,(1+x^(k+1)*A^k)^2/(1-x^(k+1)*(A+x*O(x^n))^k)^2));polcoeff(A,n)}

{a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,n,x^m*prod(k=0,m-1,(1+x^k*A^k)^2/((1-x^(k+1)*A^k +x*O(x^n))*(1-x^(k+1)*A^(k+1))))));polcoeff(A,n)}

G.f. satisfies: A(x) = 1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (1 + x^k*A(x)^k)^2/((1 - x^(k+1)*A(x)^k)*(1 - x^(k+1)*A(x)^(k+1)) due to the Heine identity.
Self-convolution of A192623.
a(n) ~ c * d^n / n^(3/2), where d = 6.65133046938958271... and c = 1.095759838870545... - Vaclav Kotesovec, Jun 30 2025

A192620 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^nA(x))^2/(1 - x^nA(x))^2.

Original entry on oeis.org

1, 4, 28, 224, 1948, 17928, 171776, 1695872, 17133436, 176297668, 1841222776, 19467629120, 207978652416, 2241618514120, 24345336854400, 266168049520832, 2927074607294300, 32356419163487336, 359330087240388828, 4007079691584624576

Offset: 0

Paul D. Hanna, Jul 06 2011

nonn

Related q-series (Heine) identity:

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

			G.f.: A(x) = 1 + 4*x + 28*x^2 + 224*x^3 + 1948*x^4 + 17928*x^5 + ...
The g.f. A = A(x) satisfies the following relations:
(0) A = (1+x*A)^2/(1-x*A)^2 * (1+x^2*A)^2/(1-x^2*A)^2 * (1+x^3*A)^2/(1-x^3*A)^2 * ...
(1) A = 1 + 4*x*A/((1-x*A)*(1-x)) + 4*x^2*A^2*(1+x)^2/((1-x*A)*(1-x^2*A)*(1-x)*(1-x^2)) + 4*x^3*A^3*(1+x)^2*(1+x^2)^2/((1-x*A)*(1-x^2*A)*(1-x^3*A)*(1-x)*(1-x^2)*(1-x^3)) + ...
(2) A^(1/2) = 1 + 2*x*A/(1-x) + 2*x^2*A^2*(1+x)/((1-x)*(1-x^2)) + 2*x^3*A^3*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3)) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A190862, A192621, A192622, A192623, A192624, A192625.

nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x])^2/(1 - x^k*A[x])^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Oct 04 2023 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, Sqrt[(r*s^(3/2)*((-1 + s)*Derivative[0, 1][QPochhammer][-s, r] + Sqrt[s]*(1 + s)*Derivative[0, 1][QPochhammer][s, r]))/(2* Pi*(1 + s)*QPochhammer[s, r]* (2* s*((1 + s^2)/(-1 + s^2)^2) + (QPolyGamma[1, Log[-s]/Log[r], r] - QPolyGamma[1, Log[s]/Log[r], r])/ Log[r]^2))]} /. FindRoot[{(-1 + s)^2*(QPochhammer[-s, r]^2/((1 + s)^2*QPochhammer[s, r]^2)) == s, 1 - 4*(s/(-1 + s^2)) + (2*(QPolyGamma[0, Log[-s]/Log[r], r] - QPolyGamma[0, Log[s]/Log[r], r]))/Log[r] == 0}, {r, 1/12}, {s, 2}, WorkingPrecision -> 120]] (* Vaclav Kotesovec, Mar 03 2024 *)

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

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

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

G.f. A(x) satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^n * Product_{k=1..n} (1+x^(k-1))^2/((1-x^k*A(x))*(1-x^k), due to the Heine identity.
(2) A(x)^(1/2) = 1 + Sum_{n>=1} x^n*A(x)^n * Product_{k=1..n} (1+x^(k-1))/(1-x^k), due to the q-binomial theorem.
Equals the self-convolution of A192621.
a(n) ~ c * d^n / n^(3/2), where d = 12.042513458183758627924432194393539477581... and c = 1.04958502741924123967536156787764354342367951743839... - Vaclav Kotesovec, Oct 04 2023
Radius of convergence r = 0.083039143238027913107320323917684421045... = 1/d and A(r) = 2.638555772492608872250287025192536127217... satisfy A(r) = 1 / Sum_{n>=1} 4*r^n/(1 - r^(2*n)*A(r)^2) and A(r) = Product_{n>=1} (1 + r^n*A(r))^2/(1 - r^n*A(r))^2. - Paul D. Hanna, Mar 02 2024

A192621 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^nA(x)^2)/(1 - x^nA(x)^2).

Original entry on oeis.org

1, 2, 12, 88, 726, 6456, 60392, 585792, 5838764, 59440250, 615431464, 6460681656, 68607630680, 735682014648, 7954732578032, 86635206695808, 949518438959574, 10464751843723840, 115904823140622164, 1289419736206548408, 14401729960605163272

Offset: 0

Paul D. Hanna, Jul 06 2011

nonn

Related q-series (Heine) identity:

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

			G.f.: A(x) = 1 + 2*x + 12*x^2 + 88*x^3 + 726*x^4 + 6456*x^5 + ...
The g.f. A = A(x) satisfies the following relations:
(0) A = (1+x*A^2)/(1-x*A^2) * (1+x^2*A^2)/(1-x^2*A^2) * (1+x^3*A^2)/(1-x^3*A^2) * ...
(1) A = 1 + 2*x*A^2/(1-x) + 2*x^2*A^4*(1+x)/((1-x)*(1-x^2)) + 2*x^3*A^6*(1+x)*(1+x^2)/((1-x)*(1-x^2)*(1-x^3)) + ...
(2) A = 1 + 2*x*A^2/((1-x*A^2)*(1-x)) + 2*x^3*A^4*(1+x)/((1-x*A^2)*(1-x^2*A^2)*(1-x)*(1-x^2)) + 2*x^6*A^6*(1+x)*(1+x^2)/((1-x*A^2)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x)*(1-x^2)*(1-x^3)) + ...
(3) A^2 = 1 + 4*x*A^2/((1-x*A^2)*(1-x)) + 4*x^2*A^4*(1+x)^2/((1-x*A^2)*(1-x^2*A^2)*(1-x)*(1-x^2)) + 4*x^3*A^6*(1+x)^2*(1+x^2)^2/((1-x*A^2)*(1-x^2*A^2)*(1-x^3*A^2)*(1-x)*(1-x^2)*(1-x^3)) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Cf. A192620 (g.f. A(x)^2), A192623, A190862, A196150, A196151.

nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x]^2)/(1 - x^k*A[x]^2), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Oct 04 2023 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{QPochhammer[-s^2, r] / QPochhammer[s^2, r] == s*((1 + s^2)/(1 - s^2)), QPolyGamma[0, Log[-s^2]/Log[r], r] - QPolyGamma[0, Log[s^2]/Log[r], r] == (2*(s^2/(s^4 - 1)) - 1/2) * Log[r]}, {r, 1/12}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 04 2023 *)

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

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

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

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

G.f. A(x) satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))/(1-x^k), due to the q-binomial theorem.
(2) A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))/((1-x^k*A(x)^2)*(1-x^k)), due to the Heine identity.
(3) A(x)^2 = 1 + Sum_{n>=1} x^n*A(x)^(2*n) * Product_{k=1..n} (1+x^(k-1))^2/((1-x^k*A(x)^2)*(1-x^k), due to the Heine identity.
Self-convolution yields A192620.
a(n) ~ c * d^n / n^(3/2), where d = 12.042513458183758627924432194393539477581... and c = 0.323075847195701225672585138139173170517867693... - Vaclav Kotesovec, Oct 04 2023
Radius of convergence r = 0.083039143238027913107320323917684421045... = 1/d and A(r) = 1.624363189835514855585723923742556266289... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 4*r^n/(1 - r^(2*n)*A(r)^4) ) and A(r) = Product_{n>=1} (1 + r^n*A(r)^2)/(1 - r^n*A(r)^2). - Paul D. Hanna, Mar 02 2024

A192627 G.f. satisfies: A(x) = Product_{n>=0} (1 + x(x+x^2)^n)/(1 - x(x+x^2)^n).

Original entry on oeis.org

1, 2, 4, 10, 22, 48, 104, 222, 466, 966, 1988, 4060, 8222, 16528, 33024, 65620, 129698, 255096, 499508, 974032, 1891866, 3661034, 7060324, 13572010, 26009822, 49701946, 94714606, 180022550, 341316642, 645594510, 1218377230, 2294387492, 4311757732

Offset: 0

Paul D. Hanna, Jul 16 2011

nonn

Related q-series identity due to Heine:
1 + Sum_{n>=1} x^n*Product_{k=0..n-1} (y+q^k)*(z+q^k)/((1-x*q^k)*(1-q^(k+1)) = Product_{n>=0} (1+x*y*q^n)*(1+x*z*q^n)/((1-x*q^n)*(1-x*y*z*q^n));
here q=x+x^2, x=x, y=1, z=0.

			G.f.: A(x) = 1 + 2*x + 4*x^2 + 10*x^3 + 22*x^4 + 48*x^5 + 104*x^6 +...
where the g.f. equals the product:
A(x) = (1+x)/(1-x) * (1+x^2*(1+x))/(1-x^2*(1+x)) * (1+x^3*(1+x)^2)/(1-x^3*(1+x)^2) * (1+x^4*(1+x)^3)/(1-x^4*(1+x)^3) *...
which is also equal to the sum:
A(x) = 1 + 2*x/((1-x)*(1-x*(1+x))) + 2*x^3*(1+x)*(1+x*(1+x))/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)) + 2*x^6*(1+x)*(1+x*(1+x))*(1 + x^2*(1+x)^2)^2/((1-x)*(1-x*(1+x))*(1-x^2*(1+x))*(1-x^2*(1+x)^2)*(1-x^3*(1+x)^2)*(1-x^3*(1+x)^3)) +...

Cf. A192626, A192623.

{a(n)=local(A=1+x);A=prod(k=0,n+1,(1+x*(x+x^2)^k)/(1-x*(x+x^2+x*O(x^n))^k));polcoeff(A,n)}

{a(n)=local(A=1+x);A=1+sum(m=1,n,x^m*(x+x^2)^(m*(m-1)/2)*prod(k=0,m-1,(1+(x+x^2)^k)/((1-x*(x+x^2)^k +x*O(x^n))*(1-(x+x^2)^(k+1)))));polcoeff(A,n)}

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

Self-convolution equals A192626.

cp's OEIS Frontend

A192622 G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)*A(x)^n)^2/(1 - x^(n+1)*A(x)^n)^2.

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A192620 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))^2/(1 - x^n*A(x))^2.

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A192621 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x)^2)/(1 - x^n*A(x)^2).

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A192627 G.f. satisfies: A(x) = Product_{n>=0} (1 + x*(x+x^2)^n)/(1 - x*(x+x^2)^n).

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A192622 G.f. satisfies: A(x) = Product_{n>=0} (1 + x^(n+1)A(x)^n)^2/(1 - x^(n+1)A(x)^n)^2.

A192620 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^nA(x))^2/(1 - x^nA(x))^2.

A192621 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^nA(x)^2)/(1 - x^nA(x)^2).

A192627 G.f. satisfies: A(x) = Product_{n>=0} (1 + x(x+x^2)^n)/(1 - x(x+x^2)^n).