A196150 -id:A196150 - cp's OEIS frontend

A145268 G.f. A(x) satisfies A(x) = 1/Product_{k>0} (1-x^k*A(x)).

1, 1, 3, 9, 30, 104, 378, 1414, 5424, 21208, 84244, 339008, 1379173, 5663078, 23439651, 97692524, 409650348, 1727034770, 7315915371, 31124324364, 132926220818, 569695276362, 2449395461726, 10561857055472, 45664873651576

Offset: 0

Vladeta Jovovic, Oct 05 2008

			G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 30*x^4 + 104*x^5 + 378*x^6 +...
The g.f. satisfies:
(0) A(x) = 1/((1-x*A(x)) * (1-x^2*A(x)) * (1-x^3*A(x)) *...).
(1) A(x) = 1 + x*A(x)/(1-x) + x^2*A(x)^2/((1-x)*(1-x^2)) + x^3*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) +...
(2) A(x) = 1 + x*A(x)/[(1-x)*(1-x*A(x))] + x^4*A(x)^2/[(1-x)*(1-x^2)*(1-x*A(x))*(1-x^2*A(x))] + x^9*A(x)^3/[(1-x)*(1-x^2)*(1-x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))] +...
(3) A(x) = 1 + x*A(x)/(1-x*A(x)) + x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
(4) A(x) = exp( x*A(x)/(1-x) + x^2*A(x)^2/(2*(1-x^2)) + x^3*A(x)^3/(3*(1-x^3)) +...).

Vaclav Kotesovec, Table of n, a(n) for n = 0..250

Cf. A210043, A145267, A196150.

terms = 25; A[] = 0; Do[A[x] = 1/Product[1-x^k A[x], {k, 1, j}] + O[x]^j, {j, 1, terms}]; CoefficientList[A[x], x] (* Jean-François Alcover, Jan 15 2018 *)
(* Calculation of constants {d,c}: *) {1/r, Sqrt[(r*(s-1)*s^3*Derivative[0, 1][QPochhammer][s, r]) / (2*Pi*((s-1)^2 * (QPolyGamma[1, Log[s]/Log[r], r]/Log[r]^2) - s))]} /. FindRoot[{s*QPochhammer[s, r] == 1 - s, 1 + s/(1 - s) == (Log[1 - r] + QPolyGamma[0, Log[s]/Log[r], r])/Log[r]}, {r, 1/5}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 28 2023 *)

{a(n)=local(A=1+x);for(i=1,n,A=1/prod(k=1,n,(1-x^k*A+x*O(x^n))));polcoeff(A,n)} /* Paul D. Hanna, May 21 2011 */

{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+x*O(x^n)))));polcoeff(A,n)} /* Paul D. Hanna, May 21 2011 */

{a(n)=local(A=1+x);for(i=1,n,A=1+sum(m=1,sqrtint(n+1),x^(m^2)*A^m/prod(k=1,m,(1-x^k)*(1-x^k*A+x*O(x^n)))));polcoeff(A,n)} /* Paul D. Hanna, May 21 2011 */

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, x^m*A/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)} /* Paul D. Hanna, Feb 11 2012 */

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1,n,x^m*A^m/(m*(1-x^m +x*O(x^n))))));polcoeff(A,n)} /* Paul D. Hanna, Mar 16 2012 */

G.f. A(x) satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^n / Product_{k=1..n} (1-x^k) due to an identity of Euler. - Paul D. Hanna, May 21 2011
(2) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^n / [Product_{k=1..n} (1-x^k)*(1-x^k*A(x))] due to Cauchy's identity. - Paul D. Hanna, May 21 2011
(3) A(x) = 1 + Sum_{n>=1} x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)) due to an identity of Euler. - Paul D. Hanna, Feb 11 2012
(4) A(x) = exp( Sum_{n>=1} x^n*A(x)^n / (n*(1-x^n)) ). - Paul D. Hanna, Mar 16 2012
a(n) ~ c * d^n / n^(3/2), where d = 4.6001032462748928128832068474594... and c = 0.695157167276255862302452181... - Vaclav Kotesovec, Aug 12 2021
Radius of convergence r = 0.2173864251437807911560951549077... = 1/d and A(r) = 2.126717513863405832814236571639... satisfy (a) A(r) = 1 / Sum_{n>=1} r^n/(1 - r^n*A(r)) and (b) A(r) = 1 / Product_{n>=1} (1 - r^n*A(r)). - Paul D. Hanna, Mar 02 2024

More terms from Max Alekseyev, Jan 31 2010

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

A196151 G.f. satisfies A(x) = Product_{n>=1} (1 + x^n*A(x)^2).

Original entry on oeis.org

1, 1, 3, 11, 43, 179, 778, 3491, 16051, 75235, 358170, 1727124, 8418266, 41408344, 205289265, 1024737905, 5145933602, 25978844478, 131773584768, 671239285119, 3432304205872, 17611565623950, 90652384728648, 467963720803022, 2422110238147351

Offset: 0

Paul D. Hanna, Sep 28 2011

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 11*x^3 + 43*x^4 + 179*x^5 + 778*x^6 + ...
where
(0) A(x) = (1+x*A(x)^2) * (1+x^2*A(x)^2) * (1+x^3*A(x)^2) * (1+x^4*A(x)^2) * ...
(1) A(x) = 1 + x*A(x)^2/(1-x) + x^3*A(x)^4/((1-x)*(1-x^2)) + x^6*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + x^10*A(x)^8/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) + ...
(2) A(x) = (1+x*A(x)^2) + x^2*A(x)^2*(1 + x^3*A(x)^2)*(1+x*A(x)^2)/(1-x) + x^7*A(x)^4*(1 + x^5*A(x)^2)*(1+x*A(x)^2)*(1+x^2*A(x)^2)/((1-x)*(1-x^2)) + x^15*A(x)^6*(1 + x^7*A(x)^2)*(1+x*A(x)^2)*(1+x^2*A(x)^2)*(1+x^3*A(x)^2)/((1-x)*(1-x^2)*(1-x^3)) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..230

Cf. A196150, A145267, A145268, A190822.

nmax = 30; A[] = 0; Do[A[x] = Product[1 + x^k*A[x]^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 28 2023 *)
(* Calculation of constants {d,c}: *) {1/r, Sqrt[-r*s*(1 + s^2) * Derivative[0, 1][QPochhammer][-s^2, r] / (8*Pi*(s^2 + (1 + s^2)^2 * QPolyGamma[1, Log[-s^2]/Log[r], r]/ Log[r]^2))]} /. FindRoot[{QPochhammer[-s^2, r] == s*(1 + s^2), 1/2 + s^2/(1 + s^2) + (Log[1 - r] + QPolyGamma[0, Log[-s^2]/Log[r], r])/Log[r] == 0}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 28 2023 *)

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

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

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

G.f. satisfies:
(1) A(x) = Sum_{n>=0} x^(n*(n+1)/2)*A(x)^(2*n) / Product_{k=1..n} (1-x^k).
(2) A(x) = Sum_{n>=0} x^(n*(3n+1)/2)*A(x)^(2*n)*(1 + x^(2n+1)*A(x)^2)*Product_{k=1..n} (1 + x^k*A(x)^2)/(1-x^k) due to Sylvester's identity.
a(n) ~ c * d^n / n^(3/2), where d = 5.5051727555189932106045782067309509... and c = 0.4987046473347092789085107139372... - Vaclav Kotesovec, Sep 28 2023
Radius of convergence r = 0.181647342310464199522927295317... = 1/d and A(r) = 1.82512871645978495662055342941... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 2*r^n/(1 + r^n*A(r)^2) ) and A(r) = Product_{n>=1} (1 + r^n*A(r)^2). - Paul D. Hanna, Mar 03 2024

A210043 G.f. A(x) satisfies: A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^(n-1)).

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 73, 211, 629, 1912, 5913, 18531, 58739, 187963, 606416, 1970326, 6441623, 21175056, 69946082, 232054411, 772886274, 2583325555, 8662455004, 29132638803, 98240253058, 332105822674, 1125273780474, 3820859749502, 12999287203624

Offset: 0

Paul D. Hanna, Mar 16 2012

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 10*x^4 + 26*x^5 + 73*x^6 + 211*x^7 +...
The g.f. satisfies the q-series identities:
(0) A(x) = 1/( (1-x) * (1-x^2*A(x)) * (1-x^3*A(x)^2) * (1-x^4*A(x)^3) *...).
(1) A(x) = 1 + x/(1-x*A(x)) + x^2/((1-x*A(x))*(1-x^2*A(x)^2)) + x^3/((1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)) +...
(2) A(x) = 1 + x/(1-x) + x^2*A(x)/((1-x)*(1-x^2*A(x))) + x^3*A(x)^2/((1-x)*(1-x^2*A(x))*(1-x^3*A(x)^2)) +...
(3) A(x) = 1 + x/((1-x)*(1-x*A(x))) + x^4*A(x)^2/((1-x)*(1-x^2*A(x))*(1-x*A(x))*(1-x^2*A(x)^2)) + x^9*A(x)^6/((1-x)*(1-x^2*A(x))*(1-x^3*A(x)^2)*(1-x*A(x))*(1-x^2*A(x)^2)*(1-x^3*A(x)^3)) +...
(4) A(x) = exp( x/(1-x*A(x)) + x^2/(2*(1-x^2*A(x)^2)) + x^3/(3*(1-x^3*A(x)^3)) +...).

Vaclav Kotesovec, Table of n, a(n) for n = 0..325 (terms 0..200 from Paul D. Hanna)

Cf. A145268, A145267, A196150, A109085.

nmax = 30; A[] = 0; Do[A[x] = 1/(1 - x)/Product[1 - x^k*A[x]^(k - 1), {k, 2, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 28 2023 *)
(* Calculation of constants {d,c}: *) {1/r, -s*Log[r*s]* Sqrt[(-1 + r*s)*(((-2 + s)*Log[r*s] + (-1 + s)*Log[1 - r*s] + (-1 + s)*QPolyGamma[0, Log[1/s]/Log[r*s], r*s])/ (2* Pi*(Log[r*s]*(4*r*(-1 + s)*s*ArcTanh[1 - 2*r*s] + 2*(-3 + s)*(-1 + r*s)*Log[r*s]^2 + (2 - 2*s + (-5 + 3*s)*(-1 + r*s)*Log[r*s])* Log[1 - r*s] + (-1 + s)*(-1 + r*s)*Log[1 - r*s]^2) + (-1 + r*s)* Log[r*s]*((-5 + 3*s)*Log[r*s] + 2*(-1 + s)*(1 + Log[1 - r*s]))* QPolyGamma[0, Log[1/s]/Log[r*s], r*s] + (-1 + s)*(-1 + r*s)*Log[r*s]* QPolyGamma[0, Log[1/s]/Log[r*s], r*s]^2 + (-1 + r*s)*((-1 + s)*(2*Log[1/s] + Log[r*s])* QPolyGamma[1, Log[1/s]/Log[r*s], r*s] + r*s*Log[r*s]^2*((-r)*s^3*Log[r*s]* Derivative[0, 2][QPochhammer][1/s, r*s] - 2*(-1 + s)* Derivative[0, 0, 1][QPolyGamma][0, Log[1/s]/Log[r*s], r*s])))))]} /. FindRoot[{s - 1 == s^2*QPochhammer[1/s, r*s], (s - 2)/s + ((s - 1)*(Log[1 - r*s] + QPolyGamma[0, Log[1/s]/Log[r*s], r*s]))/(s*Log[r*s]) + r*s^2*Derivative[0, 1][QPochhammer][1/s, r*s] == 0}, {r, 1/4}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 28 2023 *)

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

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

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

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

{a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, x^m/m/(1-x^m*A^m +x*O(x^n))))); polcoeff(A, n)}
for(n=0,35,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n / Product_{k=1..n} (1 - x^k*A(x)^k).
(2) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(n-1) / Product_{k=1..n} (1 - x^k*A(x)^(k-1)).
(3) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^(n^2-n) / [Product_{k=1..n} (1 - x^k*A(x)^(k-1))*(1 - x^k*A(x)^k)].
(4) A(x) = exp( Sum_{n>=1} x^n/n / (1 - x^n*A(x)^n) ).
a(n) ~ c * d^n / n^(3/2), where d = 3.58867546756663411130633387... and c = 0.57644814981246742030509... - Vaclav Kotesovec, Aug 12 2021

A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^nA(x)^n x^(n^2) / Product_{k=1..n} (1 - 3x^k)(1 - x^k*A(x)).

Original entry on oeis.org

1, 3, 21, 147, 1074, 8076, 62454, 494292, 3990378, 32756142, 272715870, 2297982828, 19563641319, 168036314862, 1454458825605, 12674387617266, 111104771086812, 979101922849230, 8668964794053837, 77080072176742422, 687976906966730076, 6161811541538326680

Offset: 0

Paul D. Hanna, Feb 10 2012

nonn

			G.f.: A(x) = 1 + 3*x + 21*x^2 + 147*x^3 + 1074*x^4 + 8076*x^5 +...
where the g.f. satisfies:
(0) A(x) = 1 + 3*x*A(x)/((1-3*x)*(1-x*A(x))) + 9*x^4*A(x)^2/((1-3*x)*(1-3*x^2)*(1-x*A(x))*(1-x^2*A(x))) + 27*x^9*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
(1) A(x) = 1 + 3*x*A(x)/(1-3*x) + 3*x^2*A(x)^2/((1-3*x)*(1-3*x^2)) + 3*x^3*A(x)^3/((1-3*x)*(1-3*x^2)*(1-3*x^3)) +...
(2) A(x) = 1 + 3*x*A(x)/(1-x*A(x)) + 9*x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + 27*x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...

Cf. A145268, A196150, A206637.

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*3^m*A^m/prod(k=1, m, (1-3*x^k)*(1-x^k*A+x*O(x^n))))); polcoeff(A, n)}

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, n, 3*x^m*A^m/prod(k=1, m, (1-3*x^k+x*O(x^n))))); polcoeff(A, n)}

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

G.f. satisfies the identities:
(1) A(x) = 1 + Sum_{n>=1} 3*x^n*A(x)^n / Product_{k=1..n} (1 - 3*x^k).
(2) A(x) = 1 + Sum_{n>=1} 3^n*x^n*A(x) / Product_{k=1..n} (1 - x^k*A(x)).

A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2n) / Product_{k=1..n} (1 - x^kA(x))^2.

Original entry on oeis.org

1, 1, 4, 18, 91, 489, 2751, 15985, 95218, 578324, 3568084, 22299964, 140885754, 898292262, 5772951668, 37355908797, 243184468271, 1591567315702, 10465836784159, 69114490893596, 458171948148640, 3047865264442504, 20339282134624054, 136122586785459512

Offset: 0

Paul D. Hanna, Feb 11 2012

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 91*x^4 + 489*x^5 + 2751*x^6 +...
where the g.f. satisfies:
(0) A(x) = 1 + x*A(x)^2/(1-x*A(x))^2 + x^4*A(x)^4/((1-x*A(x))^2*(1-x^2*A(x))^2) + x^9*A(x)^6/((1-x*A(x))^2*(1-x^2*A(x))^2*(1-x^3*A(x))^2) +...
(1) A(x) = 1 + x*A(x)^2/(1-x*A(x)) + x^2*A(x)^3/((1-x*A(x))*(1-x^2*A(x))) + x^3*A(x)^4/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...

Cf. A196150, A145268, A206637.

{a(n)=local(A=1+x); for(i=1, n, A=1+sum(m=1, sqrtint(n+1), x^(m^2)*A^(2*m)/prod(k=1, m, 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+1)/prod(k=1, m, (1-x^k*A+x*O(x^n))))); polcoeff(A, n)}
for(n=0,30,print1(a(n),", "))

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

cp's OEIS Frontend

A145268 G.f. A(x) satisfies A(x) = 1/Product_{k>0} (1-x^k*A(x)).

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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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A196151 G.f. satisfies A(x) = Product_{n>=1} (1 + x^n*A(x)^2).

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A210043 G.f. A(x) satisfies: A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^(n-1)).

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A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^n*A(x)^n * x^(n^2) / Product_{k=1..n} (1 - 3*x^k)*(1 - x^k*A(x)).

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A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2*n) / Product_{k=1..n} (1 - x^k*A(x))^2.

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

A206638 G.f. satisfies: A(x) = Sum_{n>=0} 3^nA(x)^n x^(n^2) / Product_{k=1..n} (1 - 3x^k)(1 - x^k*A(x)).

A206639 G.f. A(x) satisfies A(x) = Sum_{n>=0} x^(n^2) * A(x)^(2n) / Product_{k=1..n} (1 - x^kA(x))^2.