A109085 -id:A109085 - cp's OEIS frontend

A366022 Decimal expansion of a constant related to the asymptotics of A109085.

4, 8, 9, 6, 3, 5, 2, 2, 6, 6, 8, 4, 3, 0, 3, 3, 7, 3, 0, 8, 1, 5, 4, 1, 6, 6, 0, 5, 7, 8, 4, 6, 8, 6, 1, 9, 3, 2, 2, 4, 1, 6, 6, 2, 5, 1, 0, 1, 1, 5, 8, 7, 8, 4, 5, 4, 9, 4, 0, 6, 7, 2, 9, 9, 7, 0, 5, 7, 5, 8, 4, 1, 5, 7, 1, 4, 0, 1, 6, 8, 3, 2, 8, 8, 7, 0, 5, 2, 2, 9, 0, 1, 9, 6, 3, 9, 3, 8, 9, 9, 1, 7, 3, 2, 7, 6

Offset: 0

Vaclav Kotesovec, Sep 26 2023

			0.489635226684303373081541660578468619322416625...

Cf. A109085, A270915, A366018.

val = -s*Log[r*s] / Sqrt[2*Pi*((-2 - 3*Log[r*s] + 2*Log[1 - r*s])* QPolyGamma[0, 1, r*s] + QPolyGamma[0, 1, r*s]^2 - 4*ArcTanh[1 - 2*r*s]*(Log[r*s] - Log[1 - r*s]/2 - r*(s/(1 - r*s))) - 2*(Log[1 - r*s]/(1 - r*s)) - QPolyGamma[1, 1, r*s] + r*s*Log[r* s]*((-r)*s^2*Log[r*s]* Derivative[0, 2][QPochhammer][r*s, r*s] + 2*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))] /. FindRoot[{s == 1/QPochhammer[r*s], 1/s + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s])/(s* Log[r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 1000]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]]

Equals limit_{n->infinity} A109085(n) * n^(3/2) / A270915^n.

A008485 Coefficient of x^n in Product_{k>=1} 1/(1-x^k)^n.

Original entry on oeis.org

1, 1, 5, 22, 105, 506, 2492, 12405, 62337, 315445, 1605340, 8207563, 42124380, 216903064, 1119974875, 5796944357, 30068145905, 156250892610, 813310723925, 4239676354650, 22130265931900, 115654632452535, 605081974091875, 3168828466966388, 16610409114771900

Offset: 0

T. Forbes (anthony.d.forbes(AT)googlemail.com)

nonn

Number of partitions of n into parts of n kinds. - Vladeta Jovovic, Sep 08 2002
Main diagonal of A144064. - Omar E. Pol, Jun 27 2012
From Peter Bala, Apr 18 2023: (Start)
The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for all primes p and all positive integers n and k.
Conjecture: the supercongruence a(p) == p + 1 (mod p^2) holds for all primes p >= 3. Cf. A270913. (End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A000712, A000716, A023003-A023021, A005758, A006922.

Cf. A109085, A192435, A252782, A255526, A255672, A270913, A270915, A270919.

with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; `if`(n=0, 1, add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n) end end: a:= n-> etr(j->n)(n): seq(a(n), n=0..30); # Alois P. Heinz, Sep 09 2008

a[n_] := SeriesCoefficient[ Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Feb 24 2015 *)
Table[SeriesCoefficient[1/QPochhammer[x, x]^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2016 *)
Table[SeriesCoefficient[Exp[n*Sum[x^j/(j*(1-x^j)), {j, 1, n}]], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, May 19 2018 *)

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k +x*O(x^n))^n),n)}

{a(n)=n*polcoeff(log(1/x*serreverse(x*eta(x+x*O(x^n)))), n)} /* Paul D. Hanna, Apr 05 2012 */

a(n) = Sum_{pi} Product_{i=1..n} binomial(k_i+n-1, k_i) where pi runs through all nonnegative solutions of k_1+2*k_2+...+n*k_n=n. a(n) = b(n, n) where b(n, m)= m/n*Sum_{i=1..n} sigma(i)*b(n-i, m) is recurrence for number of partitions of n into parts of m kinds. - Vladeta Jovovic, Sep 08 2002
Equals the logarithmic derivative of A109085, the g.f. of which is (1/x)*Series_Reversion(x*eta(x)). - Paul D. Hanna, Apr 05 2012
Let G(x) = exp( Sum_{n>=1} a(n)*x^n/n ), then G(x) = 1/Product_{n>=1} (1-x^n*G(x)^n) is the g.f. of A109085. - Paul D. Hanna, Apr 05 2012
a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.352701333486642687772415814165..., c = A327279 = 0.26801521271073331568695383828... . - Vaclav Kotesovec, Sep 10 2014

a(0)=1 prepended by Alois P. Heinz, Mar 30 2015

A270915 Decimal expansion of a constant related to the asymptotics of A008485.

Original entry on oeis.org

5, 3, 5, 2, 7, 0, 1, 3, 3, 3, 4, 8, 6, 6, 4, 2, 6, 8, 7, 7, 7, 2, 4, 1, 5, 8, 1, 4, 1, 6, 5, 3, 2, 7, 8, 7, 9, 8, 5, 1, 4, 8, 3, 2, 7, 1, 2, 8, 6, 9, 4, 7, 0, 9, 7, 3, 1, 9, 6, 9, 0, 7, 5, 6, 0, 6, 4, 1, 0, 2, 1, 5, 1, 2, 6, 7, 5, 3, 1, 5, 5, 2, 2, 3, 2, 3, 4, 2, 7, 6, 4, 4, 7, 8, 8, 5, 4, 2, 2, 8, 2, 2, 8, 1, 7

Offset: 1

Vaclav Kotesovec, Mar 25 2016

			5.352701333486642687772415814165327879851483271286947097319690756...

Vaclav Kotesovec, Table of n, a(n) for n = 1..500

Cf. A008485, A109084, A109085, A206228, A270914, A303070, A304625, A327279.

RealDigits[1/r /. FindRoot[{s == 1/QPochhammer[r*s], QPochhammer[r*s] + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 120], 10, 105][[1]] (* Vaclav Kotesovec, Sep 26 2023 *)

Equals limit n->infinity A008485(n)^(1/n).

A181315 G.f. A(x) satisfies A(x) = Product_{n>=1} (1 + x^n*A(x)^n).

Original entry on oeis.org

1, 1, 2, 6, 19, 64, 227, 832, 3125, 11970, 46579, 183614, 731688, 2942673, 11928707, 48688888, 199932987, 825379993, 3423614756, 14261439594, 59635806865, 250241613688, 1053380320889, 4446989542144, 18823433444211, 79871578901283

Offset: 0

Paul D. Hanna, Oct 16 2010

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 19*x^4 + 64*x^5 + 227*x^6 +...
The g.f. A = A(x) satisfies
log(A) = x*A/(1-x^2*A^2) + (x^2/2)*A^2/(1-x^4*A^4) + (x^3/3)*A^3/(1-x^6*A^6) +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A000009, A081362, A109085, A270914, A366018, A366026.

nmax:=25: kmax:=nmax: for n from 1 to nmax+1 do A(x):=add(a(k)*x^k, k=0..kmax-1): A(x) := product((1 + x^k*A(x)^k),k=1..kmax+1): a(n-1):=coeff(A(x),x,n-1): od: seq(a(n),n=0..nmax); # Johannes W. Meijer, Jul 04 2011

InverseSeries[x QPochhammer[x, x^2] + O[x]^30][[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)

{a(n)=polcoeff(1/x*serreverse(x/prod(k=1,n+1,1+x^k+x*O(x^n))),n)}

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

G.f.: A(x) = Sum_{n>=0} A000009(n)*x^n*A(x)^n, where A000009(n) is the number of partitions of n into distinct parts.
G.f.: A(x) = (1/x)*Series_Reversion[x^(1/24)*eta(x)/eta(x^2)] (cf. A081362).
G.f. satisfies A(x) = exp( Sum_{n>=1} (x^n/n)*A(x)^n/(1 - (x*A(x))^(2*n)) ).
a(n) ~ c * d^n / n^(3/2), where d = A270914 = 4.50247674761735448773859393270078440676312875609162163346454... and c = A366018 = 0.482420439587319764659364391266849418507665645926542970519109122... - Vaclav Kotesovec, Aug 21 2018

A106336 Number of ways of writing n as the sum of n+1 triangular numbers, divided by n+1.

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 25, 64, 169, 442, 1172, 3180, 8730, 24116, 67159, 188568, 532741, 1512695, 4315996, 12369324, 35587923, 102747636, 297601382, 864525312, 2518185362, 7353088206, 21520084301, 63115752910, 185474840912, 546042990300, 1610314638958

Offset: 0

Paul D. Hanna, Apr 29 2005

nonn

Apparently: Number of Dyck n-paths with each ascent length being a triangular number. - David Scambler, May 09 2012

			G.f.: A(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 11*x^5 + 25*x^6 + 64*x^7 +...
A(x) = F(x*A(x)) where F(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + ...
The radius of convergence equals r = 0.322627632692191133... (A106335)
at which the g.f. converges to A(r) = 1.987369721184684145... (A106334).

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A106333, A106334, A106335, A106337; related: A109085.

b:= proc(n) option remember; expand(`if`(n=0, 1,
      add(`if`(issqr(8*j+1), x*b(n-j), 0), j=1..n)))
    end:
a:= n-> (p-> add(coeff(p, x, i)*binomial(1+n, i),
             i=0..n)/(n+1))(b(n)):
seq(a(n), n=0..35);  # Alois P. Heinz, Jul 31 2017

f[x_, y_, d_] := f[x, y, d] = If[x < 0 || y < x, 0, If[x == 0 && y == 0, 1, f[x-1, y, 0] + f[x, y - If[d == 0, 1, Ceiling[Sqrt[2*d]]],If[d == 0, 1, Ceiling[Sqrt[2*d]] + d]]]]; Table[f[n, n, 0], {n, 0, 30}] (* David Scambler, May 09 2012 *)

{a(n) = my(X); if(n<0,0,X=x+x*O(x^n); polcoef(eta(X^2)^(2*n+2)/eta(X)^(n+1)/(n+1),n))}

{a(n) = if(n<0,0,polcoef( sum(k=1,(sqrtint(8*n+1)+1)\2,x^((k^2-k)/2),x*O(x^n))^(n+1)/(n+1),n))}

{a(n) = my(A=1+x+x*O(x^n)); for(i=1,n, A=prod(m=1,n,(1+(x*A)^m)*(1-(x*A)^(2*m))));polcoef(A,n)} \\ Paul D. Hanna, Oct 23 2010

{a(n) = my(A=1+x); for(i=1,n, A=exp(sum(m=1,n,(x*A)^m/(1+(x*A)^m+x*O(x^n))/m)));polcoef(A,n)} \\ Paul D. Hanna, Jun 01 2011

G.f.: A(x) = (1/x) * Series_Reversion( x*eta(x)/eta(x^2)^2 ).
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = F(x*A(x)) where F(x) = Sum_{n>=0} x^(n*(n+1)/2).
(2) log(A(x)) = Sum_{n>=1} A106337(n)/n*x^n.
(3) A(x) = Product_{n>=1} (1 + (x*A(x))^n)*(1 - (x*A(x))^(2*n)). - Paul D. Hanna, Oct 23 2010
(4) A(x) = exp( Sum_{n>=1} (x^n*A(x)^n/(1 + x^n*A(x)^n))/n ). - Paul D. Hanna, Jun 01 2011
From Paul D. Hanna, Jun 11 2025: (Start)
(5) A(x)^4 = Sum_{n>=0} (2*n+1) * (x*A(x))^n / (1 - (x*A(x))^(2*n+1)).
(6) A(x^2)^2 = Sum_{n>=0} (x*A(x^2)^(1/2))^n / (1 + (x*A(x^2)^(1/2))^(2*n+1)).
(End)
a(n) ~ c / (n^(3/2) * A106335^n), where c = A366174 = 0.49833479793360342260635926402850016443069428233051290201996853498... - Vaclav Kotesovec, Oct 07 2020

Edited by Paul D. Hanna, Jun 01 2011

A278428 Series reversion of g.f. (1/2)x(-1; -x)_inf, where (a; q)_inf is the q-Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 2, 6, 17, 46, 128, 373, 1119, 3405, 10464, 32478, 101781, 321642, 1023512, 3276326, 10543100, 34088806, 110690682, 360810160, 1180195810, 3872588051, 12743937024, 42049240694, 139082885503, 461072582522, 1531697761470, 5098246648103, 17000237006441

Offset: 1

Vladimir Reshetnikov, Nov 21 2016

nonn

(1/2)*x*(-1; -x)_inf is the g.f. for A081360 shifted right.

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A081360, A109085, A171805, A181315, A255526.

InverseSeries[x QPochhammer[-1, -x]/2 + O[x]^35][[3]]
(* Calculation of constant c: *) 1/Sqrt[(4/s^2 - s*Derivative[0, 2][QPochhammer][-1, -s]/r) * Pi] /. FindRoot[{2*r == s*QPochhammer[-1, -s], 2*r == s^2*Derivative[0, 1][QPochhammer][-1, -s]}, {r, 1/3}, {s, 1/2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

a(n) ~ c * d^n / n^(3/2), where c = 0.1211369424750398272226454930396... and d = A318204 = 3.509754327949703340437273523375193698454789733931739911... - Vaclav Kotesovec, Nov 23 2016

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).

A171802 G.f. satisfies: A(x) = P(x*A(x)^2) where A(x/P(x)^2) = P(x) is the g.f. for Partition numbers (A000041).

Original entry on oeis.org

1, 1, 4, 20, 115, 714, 4669, 31671, 220800, 1572395, 11389059, 83642650, 621400794, 4661706035, 35264616260, 268700873765, 2060348179869, 15886552304352, 123102352038195, 958128272163860, 7487015421267228, 58715989507106041

Offset: 0

Paul D. Hanna, Dec 19 2009

nonn

			G.f.: A(x) = 1 + x + 4*x^2 + 20*x^3 + 115*x^4 + 714*x^5 +...
G.f. satisfies A(x/P(x)^2) = 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)^2 = x - 2*x^2 - x^3 + 2*x^4 + x^5 + 2*x^6 - 2*x^7 - 2*x^9 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..500

Cf. A171803, A171804, A171805, A109085, A000041, A304444, A366026.

nmax = 20; A[] = 0; Do[A[x] = 1/Product[1 - x^k*A[x]^(2*k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Sep 26 2023 *)
(* Calculation of constants {d,c}: *) eq = FindRoot[{s*QPochhammer[r*s^2] == 1, 1/s + 2*r*s^2*Derivative[0, 1][QPochhammer][r*s^2, r*s^2] == (2*(Log[1 - r*s^2] + QPolyGamma[0, 1, r*s^2]))/(s* Log[r*s^2])}, {r, 1/8}, {s, 1}, WorkingPrecision -> 1000]; {N[1/r /. eq, 100], val = -s* Log[r*s^2]*(Sqrt[1 - r*s^2]/ Sqrt[4*Pi*(16*r*s^2*ArcTanh[1 - 2*r*s^2] + (1 - r*s^2)*(Log[r*s^2] - 2*Log[1 - r*s^2])*(3*Log[r*s^2] - 2*Log[1 - r*s^2]) - 8*Log[1 - r*s^2] + 8*(1 - r*s^2)*(-1 + 2*ArcTanh[1 - 2*r*s^2]) * QPolyGamma[0, 1, r*s^2] + 4*(1 - r*s^2)*QPolyGamma[0, 1, r*s^2]^2 - 4*(1 - r*s^2)*(QPolyGamma[1, 1, r*s^2] + r*s^2*Log[r*s^2]*(r*s^3*Log[r*s^2]* Derivative[0, 2][QPochhammer][r*s^2, r*s^2] - 2* Derivative[0, 0, 1][QPolyGamma][0, 1, r*s^2])))]) /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Sep 26 2023 *)

a(n)=polcoeff((1/x*serreverse(x*eta(x+x*O(x^n))^2))^(1/2), n)

G.f. A(x) satisfies [Paul D. Hanna, Nov 24 2012]:
(1) A(x) = (1/x)*series_reversion(x*eta(x)^2).
(2) A(x) = 1 / Product_{n>=1} (1 - x^n*A(x)^(2*n)).
(3) A(x) = Sum_{n>=0} x^n*A(x)^(2*n) / Product_{k=1..n} (1-x^k*A(x)^(2*k)).
(4) A(x) = Sum_{n>=0} (x*A(x)^2)^(n^2) / Product_{k=1..n} (1-x^k*A(x)^(2*k))^2.
(5) A(x) = exp( Sum_{n>=1} (x^n/n) * A(x)^(2*n)/(1 - x^n*A(x)^(2*n)) ).
a(n) ~ c * d^n / n^(3/2), where d = 8.42516721063251541777601555584151410936... and c = 0.2128745515668564974075326286129891378270... - Vaclav Kotesovec, May 13 2018

A109084 G.f. A(x) satisfies: A(x) = 1/G000041(x/A(x)) where G000041(x) is the g.f. of the partition numbers A000041.

Original entry on oeis.org

1, -1, -2, -5, -17, -63, -253, -1062, -4615, -20570, -93538, -432211, -2023567, -9578815, -45767162, -220431025, -1069079067, -5216655257, -25592441875, -126157044454, -624560659184, -3103962569509, -15480272621533, -77450458331100, -388627340240958, -1955249529839424

Offset: 0

Paul D. Hanna, Jun 18 2005

sign

Note: coefficient [x^n] A(x)^n = -A000203(n) (sum of divisors of n) for n>0.

			The initial terms [x^0] through [x^n] of n-th self-convolution
are persistently small:
A^0: 1;
A^1: 1,-1;
A^2: 1,-2,-3;
A^3: 1,-3,-3,-4;
A^4: 1,-4,-2,0,-7;
A^5: 1,-5,0,5,0,-6;
A^6: 1,-6,3,10,3,6,-12;
A^7: 1,-7,7,14,0,7,0,-8;
A^8: 1,-8,12,16,-10,0,-8,8,-15;
A^9: 1,-9,18,15,-27,-9,-21,0,0,-13;

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A109085, A000041, A000203.

(* Calculation of constant c: *) val = Sqrt[r*s^5*(-1 + s/r)*(Log[r/s]^2 / (2*Pi*(2*s^3*(-s*Log[1 - r/s] + ArcTanh[1 - 2*r/s] * (2*r - (r - s)*(Log[1 - r/s] - 2*Log[r/s]))) + (r - s)*(s^3*(2 - 2*Log[1 - r/s] + 3*Log[r/s]) * QPolyGamma[0, 1, r/s] - s^3*QPolyGamma[0, 1, r/s]^2 + s^3*QPolyGamma[1, 1, r/s] + r*Log[r/s]*(r*Log[r/s] * Derivative[0, 2][QPochhammer][r/s, r/s] - 2*s^2*Derivative[0, 0, 1][QPolyGamma][0, 1, r/s])))))] /. FindRoot[{QPochhammer[r/s] == s, (Log[1 - r/s] + QPolyGamma[0, 1, r/s])/Log[r/s] == 1 + (r*Derivative[0, 1][QPochhammer][r/s, r/s])/s^2}, {r, 1/5}, {s, 1/2}, WorkingPrecision -> 1000]; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3] (* Vaclav Kotesovec, Oct 02 2023 *)

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

G.f.: A(x) = x/series_reversion(x*eta(x)). G.f.: A(x) = 1/G109085(x) where G109085(x) is g.f. of A109085.

a(n) ~ -c * d^n / n^(3/2), where d = A270915 = 5.35270133348664268777241581416... and c = 0.146705445870000769931272287955221766131167... - Vaclav Kotesovec, May 13 2018

A171803 G.f. satisfies: A(x) = P(x*A(x))^2 where A(x/P(x)^2) = P(x)^2 and P(x) is the g.f. for Partition numbers (A000041).

Original entry on oeis.org

1, 2, 9, 48, 286, 1818, 12086, 82992, 584079, 4190738, 30539814, 225426240, 1681904909, 12663614266, 96099303213, 734250983952, 5643749482600, 43610375803722, 338578974873523, 2639771240159904, 20659895819582337

Offset: 0

Paul D. Hanna, Dec 19 2009

nonn

			G.f.: A(x) = 1 + 2*x + 9*x^2 + 48*x^3 + 286*x^4 + 1818*x^5 +...
A(x/P(x)^2) = P(x)^2 where:
P(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 +...
P(x)^2 = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 110*x^7 +...

Cf. A171802, A171804, A171805, A109085, A000041, A304444.

nmax = 25; Rest[CoefficientList[InverseSeries[Series[x*Product[(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}]], x]] (* Vaclav Kotesovec, Nov 11 2017 *)
nmax = 30; A[] = 0; Do[A[x] = x/Product[(1 - A[x]^k)^2, {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x]/x, x] (* Vaclav Kotesovec, Oct 03 2023 *)
(* Calculation of constants {d,c}: *) eq = FindRoot[{r/QPochhammer[s]^2 == s, 1/s + 2*Sqrt[s/r]*Derivative[0, 1][QPochhammer][s, s] == (2*(Log[1 - s] + QPolyGamma[0, 1, s]))/(s*Log[s])}, {r, 1/8}, {s, 1/4}, WorkingPrecision -> 1200]; {N[1/r /. eq, 120], val = -s*Log[s]*Sqrt[(-1 + s)/(Pi*r*(r*(-8*s*Log[-1 + 1/s] + 4*(-1 + s)*Log[1 - s]^2 + 3*(-1 + s)*Log[s]^2 + 8*Log[1 - s]*(1 + Log[s] - s*Log[s])) + 8*r*(-1 + s)*(-1 + Log[-1 + 1/s])* QPolyGamma[0, 1, s] + 4*r*(-1 + s)*QPolyGamma[0, 1, s]^2 - 4*r*(-1 + s)*QPolyGamma[1, 1, s] - 4*Sqrt[r]*(-1 + s)*s^(5/2)*Log[s]^2* Derivative[0, 2][QPochhammer][s, s] + 8*r*(-1 + s)*s*Log[s]* Derivative[0, 0, 1][QPolyGamma][0, 1, s]))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

a(n)=polcoeff(1/x*serreverse(x*eta(x+x*O(x^n))^2), n)

G.f. satisfies: A(x) = 1/Product_{n>=1} (1 - A(x)^n)^2.
G.f.: A(x) = Series_Reversion(x*eta(x)^2) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
Self-convolution of A171802.
From Vaclav Kotesovec, Nov 11 2017: (Start)
a(n) ~ c * d^n / n^(3/2), where
d = 8.4251672106325154177760155558415141093613298032469849432733825... and
c = 0.6057593757525562292332998445991464666128350560350232598293... (End)

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