A145268 -id:A145268 - cp's OEIS frontend

A145267 G.f. satisfies A(x) = Product_{k>0} (1+x^k*A(x)).

1, 1, 2, 5, 12, 30, 77, 201, 532, 1427, 3868, 10579, 29161, 80931, 225954, 634197, 1788453, 5064877, 14398536, 41074364, 117541744, 337337862, 970704394, 2800059428, 8095161902, 23452565124, 68076579332, 197965830430

Offset: 0

Vladeta Jovovic, Oct 05 2008

nonn

			From _Paul D. Hanna_, May 20 2011: (Start)
G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 12*x^4 + 30*x^5 + 77*x^6 +...
G.f.: A(x) = (1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x))*(1+x^4*A(x))*...
G.f.: A(x) = (1+x*A(x)) + x^2*A(x)*(1 + x^3*A(x))*(1+x*A(x))/(1-x) + x^7*A(x)^2*(1 + x^5*A(x))*(1+x*A(x))*(1+x^2*A(x))/((1-x)*(1-x^2)) + x^15*A(x)^3*(1 + x^7*A(x))*(1+x*A(x))*(1+x^2*A(x))*(1+x^3*A(x))/((1-x)*(1-x^2)*(1-x^3)) +... (End)
G.f.: A(x) = 1 + x*A(x)/(1-x) + x^3*A(x)^2/((1-x)*(1-x^2)) + x^6*A(x)^3/((1-x)*(1-x^2)*(1-x^3)) + x^10*A(x)^4/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)) +... - _Paul D. Hanna_, Jul 01 2011

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

Cf. A145268, A190822.

nmax = 30; A[] = 0; Do[A[x] = Product[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[{QPochhammer[-s, r] == s*(1 + s), Log[1 - r] + ((1 + 2*s)*Log[r])/(1 + s) + QPolyGamma[0, Log[-s]/Log[r], r] == 0}, {r, 1/3}, {s, 1}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 26 2023 *)

{a(n)=local(A=1+x);for(i=1,n,A=prod(m=1,n,(1+A*x^m+x*O(x^n))));polcoeff(A,n)}  /* Paul D. Hanna, May 20 2011 */

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^(m*(3*m+1)/2)*A^m*(1 + x^(2*m+1)*A)*prod(k=1,m,(1+A*x^k)/(1-x^k+x*O(x^n)))));polcoeff(A,n)}  /* Paul D. Hanna, May 20 2011 */

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,x^(m*(m+1)/2)*A^m/prod(k=1,m,1-x^k +x*O(x^n))));polcoeff(A,n)} /* Paul D. Hanna, Jul 01 2011 */

G.f. satisfies: A(x) = Sum_{n>=0} x^(n*(3n+1)/2)*A(x)^n*(1 + x^(2n+1)*A(x))*Product_{k=1..n} (1 + x^k*A(x))/(1-x^k) due to Sylvester's identity. - Paul D. Hanna, May 20 2011
G.f. satisfies: A(x) = Sum_{n>=0} x^(n*(n+1)/2)*A(x)^n / Product_{k=1..n} (1-x^k). - Paul D. Hanna, Jul 01 2011
a(n) ~ c * d^n / n^(3/2), where d = 3.060735101304296413235... and c = 2.45762465379034328... - Vaclav Kotesovec, Aug 12 2021
Radius of convergence r = 0.32671889820646736561... = 1/d and A(r) = 3.6673575238633912689... satisfy (1) A(r) = 1 / Sum_{n>=1} r^n/(1 + r^n*A(r)) and (2) A(r) = 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

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

A301455 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x)^k)^k.

Original entry on oeis.org

1, 1, 4, 16, 74, 360, 1840, 9698, 52409, 288697, 1615275, 9153850, 52434770, 303104532, 1765920785, 10358843904, 61129390652, 362650003202, 2161590275029, 12938838382316, 77745063802045, 468760264760369, 2835272729215565, 17198394229862818, 104598950726341920, 637709136315071504

Offset: 0

Ilya Gutkovskiy, Mar 21 2018

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 16*x^3 + 74*x^4 + 360*x^5 + 1840*x^6 + 9698*x^7 + 52409*x^8 + 288697*x^9 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * ...).
log(A(x)) = x + 7*x^2/2 + 37*x^3/3 + 215*x^4/4 + 1251*x^5/5 + 7459*x^6/6 + 44885*x^7/7 + 272727*x^8/8 + ... + A255672(n)*x^n/n + ...

Cf. A000219, A001157, A109085, A145268, A255672, A301456.

G.f. A(x) satisfies: A(x) = exp(Sum_{k>=1} sigma_2(k)*x^k*A(x)^k/k).

A171804 G.f. satisfies: A(x) = P(x*A(x)^3) where A(x/P(x)^3) = P(x) is the g.f. for partition numbers (A000041).

Original entry on oeis.org

1, 1, 5, 33, 252, 2090, 18299, 166450, 1557595, 14898228, 145003996, 1431487820, 14299208690, 144262270360, 1467857359738, 15045486643137, 155208575698230, 1610201799670560, 16788969497000365, 175838914655128068

Offset: 0

Paul D. Hanna, Dec 20 2009

			From _Peter Bala_, Nov 12 2024: (Start)
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + ...
I(P(x)) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 153*x^5 + 646*x^6 + 2816*x^7 + ...
I^2(P(x)) = 1 + x + 4*x^2 + 20*x^3 + 115*x^4 + 714*x^5 + 4669*x^6 + 31671*x^7 + ...
I^3(P(x)) = 1 + x + 5*x^2 + 33*x^3 + 252*x^4 + 2090*x^5 + 18299*x^6 + 166450*x^7 + ... = the g.f. A(x). (End)

Cf. A000041, A010815, A109085, A145268, A171802, A171803, A171805.

a(n)=polcoeff((1/x*serreverse(x*eta(x+x*O(x^n))^3))^(1/3), n)

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

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

G.f. satisfies
(1) A(x) = 1/Product_{k>0} (1-x^k*A(x)^3).
(2) A(x) = Sum_{n>=0} x^n*A(x)^(3*n) / Product_{k=1..n} (1-x^k*A(x)^(3*k)).
(3) A(x) = Sum_{n>=0} x^(n^2)*A(x)^(3*n^2) / Product_{k=1..n} (1-x^k*A(x)^(3*k))^2.
G.f.: A(x) = 1 + x + 5*x^2 + 33*x^3 + 252*x^4 + 2090*x^5 + ...
G.f. satisfies A(x/P(x)^3) = P(x) where:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + ...
and x/P(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 + ...
Also, the g.f. A = A(x) satisfies:
(1) A(x) = 1/((1-x*A^3) * (1-x^2*A^6) * (1-x^3*A^9) * (1-x^4*A^12) * ...).
(2) A(x) = 1 + x*A^3/(1-x*A^3) + x^2*A^6/((1-x*A^3)*(1-x^2*A^6)) + x^3*A^9/((1-x*A^3)*(1-x^2*A^6)*(1-x^3*A^9)) + ...
(3) A(x) = 1 + x*A^3/(1-x*A^3)^2 + x^4*A^12/((1-x*A^3)*(1-x^2*A^6))^2 + x^9*A^27/((1-x*A^3)*(1-x^2*A^6)*(1-x^3*A^9))^2 + ...
From Peter Bala, Nov 12 2024: (Start)
A(x) = ( 1/x * series_reversion(x/P(x)^3) )^(1/3).
A(x) = the third iterate I^3(P(x)), where the operator I is defined by I(f(x)) = 1/x * series_reversion(x/f(x)). See the Example section. (Note that I(P(x)) is the g.f. of A109085 and I^2(P(x)) is the g.f. of A171802.) (End)

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

A206637 G.f. satisfies: A(x) = Sum_{n>=0} 2^nA(x)^n x^(n^2) / Product_{k=1..n} (1 - 2x^k)(1 - x^k*A(x)).

Original entry on oeis.org

1, 2, 10, 50, 266, 1466, 8370, 49090, 294458, 1798794, 11156074, 70069514, 444822530, 2849764698, 18401517066, 119640989514, 782575127258, 5146252178882, 34003440381186, 225635772455882, 1503017848153914, 10046960505610082, 67372689978768714, 453099298491559554

Offset: 0

Paul D. Hanna, Feb 10 2012

nonn

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 50*x^3 + 266*x^4 + 1466*x^5 + 8370*x^6 +...
where the g.f. satisfies:
(0) A(x) = 1 + 2*x*A(x)/((1-2*x)*(1-x*A(x))) + 4*x^4*A(x)^2/((1-2*x)*(1-2*x^2)*(1-x*A(x))*(1-x^2*A(x))) + 8*x^9*A(x)^3/((1-2*x)*(1-2*x^2)*(1-2*x^3)*(1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...
(1) A(x) = 1 + 2*x*A(x)/(1-2*x) + 2*x^2*A(x)^2/((1-2*x)*(1-2*x^2)) + 2*x^3*A(x)^3/((1-2*x)*(1-2*x^2)*(1-2*x^3)) +...
(2) A(x) = 1 + 2*x*A(x)/(1-x*A(x)) + 4*x^2*A(x)/((1-x*A(x))*(1-x^2*A(x))) + 8*x^3*A(x)/((1-x*A(x))*(1-x^2*A(x))*(1-x^3*A(x))) +...

Cf. A145268, A206638.

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

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.

cp's OEIS Frontend

A145267 G.f. satisfies A(x) = 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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A190862 G.f. satisfies: A(x) = Product_{n>=1} (1 + x^n*A(x))/(1 - x^n*A(x)).

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A301455 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - x^k*A(x)^k)^k.

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A171804 G.f. satisfies: A(x) = P(x*A(x)^3) where A(x/P(x)^3) = P(x) is the g.f. for partition numbers (A000041).

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

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A206637 G.f. satisfies: A(x) = Sum_{n>=0} 2^n*A(x)^n * x^(n^2) / Product_{k=1..n} (1 - 2*x^k)*(1 - x^k*A(x)).

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

A206637 G.f. satisfies: A(x) = Sum_{n>=0} 2^nA(x)^n x^(n^2) / Product_{k=1..n} (1 - 2x^k)(1 - x^k*A(x)).