A145267 -id:A145267 - 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

A196150 G.f. satisfies A(x) = 1/Product_{n>=1} (1 - x^n*A(x)^2).

Original entry on oeis.org

1, 1, 4, 18, 92, 505, 2922, 17541, 108270, 682823, 4380942, 28504466, 187636994, 1247375147, 8362420498, 56471709841, 383790966537, 2622982116829, 18016055333571, 124296340608870, 860986586024343, 5985590694574930, 41749023026002831

Offset: 0

Paul D. Hanna, Sep 28 2011

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 18*x^3 + 92*x^4 + 505*x^5 + 2922*x^6 + ...
where
(0) A(x) = 1/((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) + x^2*A(x)^4/((1-x)*(1-x^2)) + x^3*A(x)^6/((1-x)*(1-x^2)*(1-x^3)) + ...
(2) A(x) = 1 + x*A(x)^2/[(1-x)*(1-x*A(x)^2)] + x^4*A(x)^4/[(1-x)*(1-x^2)*(1-x*A(x)^2)*(1-x^2*A(x)^2)] + x^9*A(x)^6/[(1-x)*(1-x^2)*(1-x^3)*(1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)] + ...
(3) A(x) = 1 + x*A(x)^2/(1-x*A(x)^2) + x^2*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)) + x^3*A(x)^2/((1-x*A(x)^2)*(1-x^2*A(x)^2)*(1-x^3*A(x)^2)) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..150

Cf. A196151, A145268, A145267, A206639, A206637.

nmax = 30; A[] = 0; Do[A[x] = 1/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^3 * (s^2 - 1) * Derivative[0, 1][QPochhammer][s^2, r] / (8*Pi*((s^2 - 1)^2*(QPolyGamma[1, 2*Log[s]/Log[r], r] / Log[r]^2) - s^2))]} /. FindRoot[{(1 - s^2)/QPochhammer[s^2, r] == s, 1/2 + s^2/(1 - s^2) == (Log[1 - r] + QPolyGamma[0, 2*Log[s]/Log[r], r]) / Log[r]}, {r, 1/8}, {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^2+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^(2*m)/prod(k=1, m, (1-x^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^(2*m)/prod(k=1, m, (1-x^k)*(1-x^k*A^2+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^2/prod(k=1, m, (1-x^k*A^2+x*O(x^n))))); polcoeff(A, n)}

G.f. satisfies:
(1) A(x) = 1 + Sum_{n>=1} x^n*A(x)^(2*n) / Product_{k=1..n} (1-x^k) due to an identity of Euler.
(2) A(x) = 1 + Sum_{n>=1} x^(n^2)*A(x)^(2*n)/[Product_{k=1..n} (1-x^k)*(1-x^k*A(x)^2)] due to Cauchy's identity.
(3) A(x) = 1 + Sum_{n>=1} x^n*A(x)^2 / Product_{k=1..n} (1 - x^k*A(x)^2).
a(n) ~ c * d^n / n^(3/2), where d = 7.4702934491577480082... and c = 0.270144986991156076... - Vaclav Kotesovec, Aug 12 2021
Radius of convergence r = 0.1338635499135240586... = 1/d and A(r) = 1.5228379370493260575... satisfy A(r) = 1 / sqrt( Sum_{n>=1} 2*r^n/(1 - r^n*A(r)^2) ) and A(r) = 1 / Product_{n>=1} (1 - r^n*A(r)^2). - Paul D. Hanna, Mar 02 2024

A301456 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x)^k)^k.

Original entry on oeis.org

1, 1, 3, 12, 49, 217, 1006, 4810, 23576, 117812, 597937, 3073874, 15972678, 83758809, 442681653, 2355678968, 12610759255, 67868269712, 366979432955, 1992755590086, 10862329206524, 59414599714958, 326009477088080, 1793977307978268, 9898072238695390, 54744525395860053, 303463833091357785

Offset: 0

Ilya Gutkovskiy, Mar 21 2018

nonn

			G.f. A(x) = 1 + x + 3*x^2 + 12*x^3 + 49*x^4 + 217*x^5 + 1006*x^6 + 4810*x^7 + 23576*x^8 + 117812*x^9 + ...
G.f. A(x) satisfies: A(x) = (1 + x*A(x)) * (1 + x^2*A(x)^2)^2 * (1 + x^3*A(x)^3)^3 * (1 + x^4*A(x)^4)^4 * ...
log(A(x)) = x + 5*x^2/2 + 28*x^3/3 + 141*x^4/4 + 751*x^5/5 + 4064*x^6/6 + 22198*x^7/7 + 122381*x^8/8 + ... + A270922(n)*x^n/n + ...

Cf. A026007, A145267, A181315, A270922, A301455.

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

A190862 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^nA(x))/(1 - x^nA(x)).

Original entry on oeis.org

1, 2, 8, 36, 174, 888, 4716, 25808, 144568, 825030, 4780176, 28045860, 166295716, 994959560, 5999349896, 36420226288, 222415222446, 1365445230212, 8422174103796, 52168047039764, 324366739546304, 2023789526326096

Offset: 0

Paul D. Hanna, May 21 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=1, z=0.

			G.f.: A(x) = 1 + 2*x + 8*x^2 + 36*x^3 + 174*x^4 + 888*x^5 + ...
The g.f. A = A(x) satisfies the following relations:
(0) A = (1+x*A)/(1-x*A) * (1+x^2*A)/(1-x^2*A) * (1+x^3*A)/(1-x^3*A) * ...
(1) A = 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)) + ...
(2) A = 1 + 2*x*A/((1-x*A)*(1-x)) + 2*x^3*A^2*(1+x)/((1-x*A)*(1-x^2*A)*(1-x)*(1-x^2)) + 2*x^6*A^3*(1+x)*(1+x^2)/((1-x*A)*(1-x^2*A)*(1-x^3*A)*(1-x)*(1-x^2)*(1-x^3)) + ...
(3) A^2 = 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)) + ... (cf. A192619)

Cf. A145267, A145268, A190861, A192619 (g.f. A(x)^2), A192621.

nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x])/(1 - x^k*A[x]), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{(s-1)*QPochhammer[-s, r] == -s*(s+1) * QPochhammer[s, r], (s^2 - 1)*(QPolyGamma[0, Log[-s]/Log[r], r] - QPolyGamma[0, Log[s]/Log[r], r]) + Log[r]*(s^2 - 2*s - 1) == 0}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 26 2023 *)

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

G.f. satisfies:
(1) A(x) = 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.
(2) A(x) = 1 + Sum_{n>=1} x^(n*(n+1)/2)*A(x)^n*Product_{k=1..n} (1+x^(k-1))/((1-x^k*A(x))*(1-x^k)) due to the Heine identity.
(3) A(x)^2 = 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.
a(n) ~ c * d^n / n^(3/2), where d = 6.6934289011143535333002543297069340451347... and c = 0.946606599119645056034760125205426820822370610602636232678... - Vaclav Kotesovec, Sep 26 2023
Radius of convergence r = 0.149400257293166331446262618504038357688... = 1/d and A(r) = 2.500666835731534833961673247439001530869... satisfy A(r) = 1 / Sum_{n>=1} 2*r^n/(1 - r^(2*n)*A(r)^2) and A(r) = Product_{n>=1} (1 + r^n*A(r))/(1 - r^n*A(r)). - Paul D. Hanna, Mar 02 2024

A190822 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n) * (1 + x^(2n)*A(x)).

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 27, 53, 104, 208, 415, 836, 1690, 3434, 7004, 14342, 29460, 60707, 125443, 259883, 539689, 1123226, 2342493, 4894590, 10245321, 21481047, 45108768, 94863801, 199772929, 421245065, 889331420, 1879723964, 3977402460, 8424718846

Offset: 0

Paul D. Hanna, May 21 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 4*x^3 + 7*x^4 + 14*x^5 + 27*x^6 + ...
G.f.: A(x) = (1+x)*(1+x^2*A(x)) * (1+x^2)*(1+x^4*A(x)) * (1+x^3)*(1+x^6*A(x)) * ...
G.f.: A(x) = 1 + x*(1+x*A(x))/(1-x) + x^3*(1+x*A(x))*(1+x^2*A(x))/((1-x)*(1-x^2)) + x^6*(1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x))/((1-x)*(1-x^2)*(1-x^3)) + ...

Cf. A145267.

nmax = 40; A[] = 0; Do[A[x] = Product[(1 + x^k)*(1 + x^(2*k)*A[x]), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Mar 03 2024 *)
(* Calculation of constants {d,c}: *) Chop[{1/r, Sqrt[(-r*s*(1 + s) * Log[r^2]^2 * (s*(1 + s)*Derivative[0, 1][QPochhammer][-1, r] + r*QPochhammer[-1, r]^2 * Derivative[0, 1][QPochhammer][-s, r^2]))/(2*Pi * QPochhammer[-1, r]* (s*Log[r^2]^2 + (1 + s)^2 * QPolyGamma[1, Log[-s]/Log[r^2], r^2]))]} /. FindRoot[{2*s*(1 + s) == QPochhammer[-1, r]*QPochhammer[-s, r^2], 1 + s/(1 + s) + (Log[1 - r^2] + QPolyGamma[0, Log[-s]/Log[r^2], r^2])/Log[r^2] == 0}, {r, 1/2}, {s, 8}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Mar 03 2024 *)

{a(n) = my(A=1+x);for(i=1,n, A = prod(m=1,n, (1 + x^m) * (1 + x^(2*m)*A+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*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)) / (1 - x^k) due to a Lebesgue identity.
From Vaclav Kotesovec, Mar 03 2024: (Start)
Let A(x) = y, then 2*y*(1 + y) = QPochhammer(-1, x) * QPochhammer(-y, x^2).
a(n) ~ c * d^n / n^(3/2), where d = 2.20229791253644493239805950840417681972879718454582447550768622636671... and c = 9.92694112477002167508700773789825154871250555780774205172995613775...
Radius of convergence:
r = 1/d = 0.45407117461609608946909851977877786178200148047136427660297778018...
A(r) = s = 8.6584215712749049134273598177515922912152713325328273868580739614...
(End)
The values r and A(r) given above also satisfy A(r) = 1 / Sum_{n>=1} r^(2*n)/(1 + r^(2*n)*A(r)). - Paul D. Hanna, Mar 03 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

A298260 G.f. A(x) satisfies A(x) = Product_{k>=1} 1/(1 + x^k*A(x)).

Original entry on oeis.org

1, -1, 1, -3, 8, -22, 62, -182, 550, -1694, 5294, -16758, 53635, -173260, 564129, -1849448, 6099972, -20227036, 67390803, -225485432, 757361764, -2552692848, 8631144354, -29268108530, 99511629658, -339167845294, 1158607479710, -3966129297519, 13603228472518

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

sign

			G.f. A(x) = 1 - x + x^2 - 3*x^3 + 8*x^4 - 22*x^5 + 62*x^6 - 182*x^7 + 550*x^8 - 1694*x^9 + ...
G.f. A(x) satisfies A(x) = 1/((1 + x*A(x)) * (1 + x^2*A(x)) * (1 + x^3*A(x)) * ... ).

Cf. A145267, A145268, A298261.

A298261 G.f. A(x) satisfies A(x) = Product_{k>=1} (1 - x^k*A(x)).

Original entry on oeis.org

1, -1, 0, 1, -2, 2, -1, -1, 4, -9, 16, -19, 1, 59, -158, 229, -129, -297, 1066, -1878, 1992, -216, -4862, 13912, -24258, 25406, 4162, -90120, 233708, -359262, 264319, 360325, -1745699, 3624263, -4623550, 1795485, 8918014, -29893776, 55251854, -61018833, -1455525

Offset: 0

Ilya Gutkovskiy, Jan 15 2018

sign

			G.f. A(x) = 1 - x + x^3 - 2*x^4 + 2*x^5 - x^6 - x^7 + 4*x^8 - 9*x^9 + ...
G.f. A(x) satisfies A(x) = (1 - x*A(x)) * (1 - x^2*A(x)) * (1 - x^3*A(x)) * ...

Cf. A145267, A145268, A298260.

A302171 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x))^k.

Original entry on oeis.org

1, 1, 4, 14, 54, 213, 880, 3724, 16143, 71227, 319067, 1447160, 6633530, 30682425, 143028870, 671293632, 3169572659, 15044993968, 71752624923, 343658572717, 1652266087698, 7971518032791, 38581202763318, 187269381724629, 911404238805468, 4446493502832481, 21742327471261176

Offset: 0

Ilya Gutkovskiy, Apr 02 2018

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 14*x^3 + 54*x^4 + 213*x^5 + 880*x^6 + 3724*x^7 + 16143*x^8 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - x^2*A(x))^2 * (1 - x^3*A(x))^3 * ...).

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

Cf. A000219, A145267, A145268, A190862, A298260, A298261, A301455, A301624, A301831.

nmax = 30; A[] = 0; Do[A[x] = 1/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 = 5.177446537296361283814259811908762546749... and c = 0.81395777803098291048009263980507199... - Vaclav Kotesovec, Sep 27 2023

Radius of convergence r = 0.1931454033945844258723936803941781838... = 1/d and A(r) = 2.2252305561396523944672847657756264073... satisfy (1) A(r) = 1 / Sum_{n>=1} n*r^n/(1 - r^n*A(r)) and (2) A(r) = 1 / Product_{n>=1} (1 - r^n*A(r))^n. - Paul D. Hanna, Mar 02 2024

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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A196150 G.f. satisfies A(x) = 1/Product_{n>=1} (1 - x^n*A(x)^2).

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A301456 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 + x^k*A(x)^k)^k.

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A190862 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))/(1 - x^n*A(x)).

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A190822 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n) * (1 + x^(2n)*A(x)).

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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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A190862 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^nA(x))/(1 - x^nA(x)).