A109085 -id:A109085 - cp's OEIS frontend

A171805 G.f.: Series reversion of x/P(x)^3 where P(x) is the g.f. for Partition numbers (A000041).

1, 3, 18, 130, 1044, 8946, 80135, 741312, 7027515, 67911855, 666525630, 6625647054, 66570488901, 674964968175, 6897258376218, 70961851119848, 734455079297433, 7641851681095236, 79886815507105175, 838655487787502616, 8837797224686207976, 93454820274339167191

Offset: 1

Paul D. Hanna, Dec 20 2009

nonn

			G.f.: A(x) = x + 3*x^2 + 18*x^3 + 130*x^4 + 1044*x^5 + 8946*x^6 +...
where Series_Reversion(A(x)) = x/P(x)^3 = x*eta(x)^3 and
x*eta(x)^3 = x - 3*x^2 + 5*x^4 - 7*x^7 + 9*x^11 - 11*x^16 + 13*x^22 +...

Cf. A109085, A171802, A171803, A171804, A000041, A007312, A010815, A010816.

InverseSeries[x QPochhammer[x]^3 + O[x]^30][[3]] (* Vladimir Reshetnikov, Nov 21 2016 *)
(* Calculation of constants {d,c}: *) eq = FindRoot[{r/QPochhammer[s]^3 == s, 1/s + 3*(s/r)^(1/3)*Derivative[0, 1][QPochhammer][s, s] == (3*(Log[1 - s] + QPolyGamma[0, 1, s]))/(s*Log[s])}, {r, 1/10}, {s, 1/8}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[r*(-1 + s)*s^2*(Log[s]^2/(6*Pi*(r*(-4*s*ArcTanh[1 - 2*s] + Log[1 - s]*(2 + 3*(-1 + s)*Log[1 - s] + Log[s] - s*Log[s])) - (-1 + s)*(-3*r*QPolyGamma[0, 1, s]^2 + r*QPolyGamma[1, 1, s] + QPolyGamma[0, 1, s]*(r*(2 - 6*Log[1 - s] + Log[s]) + 6*(r/s)^(2/3)*s^2*Log[s]* Derivative[0, 1][QPochhammer][s, s]) + s*Log[s]*((r/s)^(1/3)*s*(6*(r/s)^(1/3) * Log[1 - s] * Derivative[0, 1][QPochhammer][s, s] - 4*s*Log[s] * Derivative[0, 1][QPochhammer][s, s]^2 + (r/s)^(1/3)*s*Log[s]* Derivative[0, 2][QPochhammer][s, s]) - 2*r*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(serreverse(x*eta(x+x*O(x^n))^3),n)}

G.f. A(x) satisfies:
(1) A(x) = x/Product_{n>=1} (1 - A(x)^n)^3 ;
(2) A(x) = x/Sum_{n>=0} (-1)^n*(2n+1)*A(x)^(n(n+1)/2).
G.f.: A(x) = Series_Reversion(x*eta(x)^3) where eta(q) is the q-expansion of the Dedekind eta function without the q^(1/24) factor (A010815).
Self-convolution cube of A171804 (with offset).
a(n) ~ c * d^n / n^(3/2), where d = 11.34340769381039824727582112969136186... and c = 0.05972244738388663765328174469956... - Vaclav Kotesovec, Nov 11 2017

More terms from Vladimir Reshetnikov, Nov 21 2016

A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

Original entry on oeis.org

1, -1, -1, 4, 1, -17, -6, 118, -8, -876, 625, 5966, -7486, -41937, 75969, 306312, -768637, -2164992, 7487063, 14461466, -70259884, -89410774, 646971980, 459817892, -5861484630, -1128608133, 52082250637, -15894742662, -453574650852, 366848121166, 3866670213663, -5215687717614

Offset: 0

Ilya Gutkovskiy, Mar 24 2018

sign

			G.f. A(x) = 1 - x - x^2 + 4*x^3 + x^4 - 17*x^5 - 6*x^6 + 118*x^7 - 8*x^8 - 876*x^9 + 625*x^10 + ...
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 - 3*x^2/2 + 8*x^3/3 + 13*x^4/4 - 51*x^5/5 - 120*x^6/6 + 538*x^7/7 + 781*x^8/8 - 5419*x^9/9 - 3053*x^10/10 + ... + A281267(n)*x^n/n + ...

Cf. A000219, A006195, A066398, A073592, A109085, A181315, A278428, A281267, A301455, A301456, A301625.

with(numtheory):
Order := 33:
Gser := solve(series(x*exp(add(sigma[2](n)*x^n/n, n = 1..32)), x) = y, x):
seq(coeff(Gser, y^k), k = 1..32); # Peter Bala, Feb 09 2020

From Peter Bala, Feb 09 2020: (Start)
A(x) = 1/x * series reversion of ( exp( Sum_{n >= 1} sigma_2(n)*x^n/n ) ), where sigma_2(n) = A001157(n).
Equivalently, the o.g.f. A(x) satisfies [x^n](1/A(x))^n = sigma_2(n) for n >= 1. Cf. A066398. (End)
A(x) equals (1/x) * series reversion of (x * the o.g.f. for the sequence of planar partitions A000219).  - Peter Bala, Feb 11 2020

A006195 Reversion of Jacobi theta_3.

Original entry on oeis.org

1, -2, 8, -40, 222, -1316, 8160, -52272, 343220, -2297682, 15623760, -107611608, 749209832, -5264005060, 37277153920, -265788870480, 1906489923022, -13747860118724, 99606357848920, -724732875917064, 5293303253527704, -38795196044205056

Offset: 0

N. J. A. Sloane

sign

J.-G. Penaud, Arbres et Animaux, Ph.D. Dissertation, Université Bordeaux I, 1990, cover page and p. 76. (Annotated scanned copy)
Index entries for reversions of series

Cf. A000122, A109085, A181315.

# Using function CompInv from A357588.
CompInv(22, n -> if n = 1 then 1 elif issqr(n-1) then 2 else 0 fi); # Peter Luschny, Oct 05 2022

nmax = 30; A[] = 0; Do[A[x] = Product[(1 + x^k*A[x]^k)/(1 - x^k*A[x]^k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * (-1)^Range[0, nmax] (* Vaclav Kotesovec, Sep 27 2023 *)
(* Calculation of constant d: *) 1/r /. FindRoot[{QPochhammer[-1, r*s] == 2*s*QPochhammer[r*s], (2* QPochhammer[r*s]*(-Log[r*s] + Log[1 - r*s] + QPolyGamma[0, 1, r*s])) / Log[r*s] + r*(Derivative[0, 1][QPochhammer][-1, r*s] - 2*s*Derivative[0, 1][QPochhammer][r*s, r*s]) == 0}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Sep 27 2023 *)

N=66; x='x+O('x^N); /* that many terms */
Vec(serreverse(x*sum(n=-N,N,x^(n^2)))) /* show terms */ /* Joerg Arndt, May 25 2011 */

REVERT(A000122).
From Vaclav Kotesovec, Sep 27 2023: (Start)
G.f. A(x) satisfies A(x) = Product_{k>=1} (1 + (-x)^k*A(x)^k)/(1 - (-x)^k*A(x)^k).
a(n) ~ (-1)^n * c * d^n / n^(3/2), where d = 7.86298339570590526151934790995382716105758424871057843176888470144337... and c = 0.617020565581840591336246430220953133238702598666548444780767269...
(End)

Signs corrected by N. J. A. Sloane, Dec 24 2001

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)

A184362 G.f.: eta(x) + x*eta'(x).

Original entry on oeis.org

1, -2, -3, 0, 0, 6, 0, 8, 0, 0, 0, 0, -13, 0, 0, -16, 0, 0, 0, 0, 0, 0, 23, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0, 0, -36, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 52, 0, 0, 0, 0, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -71, 0, 0, 0, 0, 0, 0, -78, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 93, 0, 0, 0

Offset: 0

Paul D. Hanna, Jan 18 2011

sign

The formulas specified in this entry use eta(x) to denote Dedekind's eta(q) function without the q^(1/24) factor.

			G.f.: A(x) = 1 - 2*x - 3*x^2 + 6*x^5 + 8*x^7 - 13*x^12 - 16*x^15 + 23*x^22 + 27*x^26 - 36*x^35 - 41*x^40 +...
Illustrate the property: [x^n] A(x)/eta(x)^(n+1) = 0
in the table of coefficients of A(x)/eta(x)^(n+1) for n=0..10:
[1, -1, -3, -4, -7, -6, -12, -8, -15, -13, -18,...,-sigma(n),...];
[1,(0), -2, -6, -15, -28, -55, -90, -154, -240, -378,...];
[1, 1,(0), -5, -20, -54, -130, -275, -555, -1050, -1924,...];
[1, 2, 3,(0), -17, -72, -221, -572, -1350, -2958, -6160,...];
[1, 3, 7, 10,(0), -63, -287, -930, -2580, -6475, -15162,...];
[1, 4, 12, 26, 38,(0), -253, -1196, -4059, -11780, -31027,...];
[1, 5, 18, 49, 105, 153,(0), -1062, -5175, -18140, -54544,...];
[1, 6, 25, 80, 210, 442, 646,(0), -4615, -22990, -82671,...];
[1, 7, 33, 120, 363, 924, 1926, 2816,(0), -20570, -104285,...];
[1, 8, 42, 170, 575, 1668, 4161, 8602, 12585,(0), -93538,...];
[1, 9, 52, 231, 858, 2756, 7766, 19071, 39182, 57343,(0),...]; ...
so that the coefficient of x^n in A(x)/eta(x)^(n+1) is zero for n>=1.
Note: the g.f.s of the diagonals in the above table are powers of G(x),
where G(x) = 1/eta(x*G(x)) is the g.f. of A109085.
The g.f. of A184363 equals:
A(x)*eta(x)^2 = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 +...+ (-1)^n*(2n+1)*(n^2+n+6)/6*x^(n(n+1)/2) +...

Cf. A184363, A184365, A109084, A109085, A000203.

{a(n)=polcoeff(eta(x+x*O(x^n)) + x*deriv(eta(x+x*O(x^n))),n)}

G.f. A(x) satisfies:
(1) [x^n] A(x)/eta(x)^(n+1) = 0 for n>=1.
(2) [x^n] A(x)/eta(x)^n = A109084(n) for n>=0.
(3) [x^n] A(x)/eta(x)^(n+2) = A109085(n) for n>=0.
(4) A(x)/eta(x) = 1 - Sum_{n>=1} sigma(n)*x^n.
(5) A(x) = 1 + Sum_{n>=1} (-1)^n*(n*(3*n-1)/2+1 + (n*(3*n+1)/2+1)*x^n) * x^(n*(3*n-1)/2).
(6) A(x)*eta(x)^2 = Sum_{n>=0} (-1)^n*(2n+1)*(n^2+n+6)/6*x^(n(n+1)/2).

Example of g.f. corrected by Paul D. Hanna, Jan 18 2011

Name changed slightly by Paul D. Hanna, Nov 27 2012

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

A238440 Expansion of 1/E(q/E(q)) where E(q) = Product_{n>=1} (1 - q^n).

Original entry on oeis.org

1, 1, 3, 9, 27, 79, 229, 657, 1873, 5304, 14944, 41895, 116947, 325133, 900617, 2486183, 6841490, 18770754, 51358188, 140154540, 381540434, 1036261537, 2808328337, 7594958401, 20499680869, 55227373266, 148520150761, 398726637407, 1068701794158, 2859956501816, 7642086948143, 20391083977989, 54333644617311

Offset: 0

Joerg Arndt, Feb 27 2014

nonn

What does this sequence count?

Cf. A109085: G.f. 1/E(q/E(q/E(q/E(q/E(q/E...))))).

q = 'q + O('q^66);  Vec( 1/eta(q/eta(q)) )

G.f.: 1/E(q/E(q)) where E(q) = Product_{n>=1} (1 - q^n).

A301577 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - kx^kA(x)^k).

Original entry on oeis.org

1, 1, 4, 16, 75, 366, 1887, 10010, 54493, 302302, 1703599, 9723774, 56101292, 326640411, 1916800425, 11325242328, 67316128903, 402245682741, 2414978550718, 14560379165160, 88122911824659, 535188028077586, 3260549998701951, 19921639754064470, 122041156818328779

Offset: 0

Ilya Gutkovskiy, Mar 23 2018

nonn

			G.f. A(x) = 1 + x + 4*x^2 + 16*x^3 + 75*x^4 + 366*x^5 + 1887*x^6 + 10010*x^7 + 54493*x^8 + 302302*x^9 + ...
G.f. A(x) satisfies: A(x) = 1/((1 - x*A(x)) * (1 - 2*x^2*A(x)^2) * (1 - 3*x^3*A(x)^3) * ...).
log(A(x)) = x + 7*x^2/2 + 37*x^3/3 + 219*x^4/4 + 1276*x^5/5 + 7687*x^6/6 + 46551*x^7/7 + 285043*x^8/8 + ... + A297329(n)*x^n/n + ...

Cf. A006906, A109085, A297329, A301455, A301578.

A192783 G.f. satisfies: A(x) = Product_{n>=1} 1/(1 - x^n*A(x)^(n^3)).

Original entry on oeis.org

1, 1, 3, 16, 119, 1145, 13301, 180464, 2821941, 50400230, 1022250876, 23424407915, 602724515761, 17299947151776, 550101222059396, 19246320555772626, 736247255316380311, 30620337253882961105, 1377609185722013042566, 66750666290443384609574

Offset: 0

Paul D. Hanna, Jul 09 2011

nonn

			G.f: A(x) = 1 + x + 3*x^2 + 16*x^3 + 119*x^4 + 1145*x^5 + 13301*x^6 +...
The g.f. A = A(x) satisfies:
A = 1/((1 - x*A)*(1 - x^2*A^8)*(1 - x^3*A^27)*(1 - x^4*A^64)*...),
as well as the logarithmic series:
log(A) = x*A + x^2*(A^2 + 2*A^8)/2 + x^3*(A^3 + 3*A^27)/3 + x^4*(A^4 + 2*A^16 + 4*A^64)/4 + x^5*(A^5 + 5*A^125)/5 + x^6*(A^6 + 2*A^24 + 3*A^54 + 6*A^216)/6  +...

Cf. A109085, A192768.

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

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

G.f. satisfies: A(x) = exp( Sum_{n>=1} (x^n/n)*Sum_{d|n} d*A(x)^(n*d^2) ).

A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^kA(x)^k)/(1 - x^kA(x)^k))^k.

Original entry on oeis.org

1, 2, 10, 60, 398, 2820, 20892, 159868, 1253758, 10024070, 81400672, 669532924, 5566386324, 46701736772, 394910202608, 3362210548344, 28797181196766, 247955463799812, 2145088563952510, 18636002388075260, 162523319555310664, 1422259430668179592, 12485554521209720492, 109922263517662775292

Offset: 0

Ilya Gutkovskiy, Mar 24 2018

nonn

			G.f. A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 398*x^4 + 2820*x^5 + 20892*x^6 + 159868*x^7 + 1253758*x^8 + ...
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*A(x)) * (1 - x^2*A(x)^2)^2 * (1 - x^3*A(x)^3)^3 * ...).
log(A(x)) = 2*x + 16*x^2/2 + 128*x^3/3 + 1056*x^4/4 + 8952*x^5/5 + 77200*x^6/6 + 673948*x^7/7 + 5937792*x^8/8 + ... + A270924(n)*x^n/n + ...

Cf. A006195, A066398, A109085, A156616, A181315, A270924, A278428, A301455, A301456, A301624.

cp's OEIS Frontend

A171805 G.f.: Series reversion of x/P(x)^3 where P(x) is the g.f. for Partition numbers (A000041).

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A301624 G.f. A(x) satisfies: A(x) = Product_{k>=1} (1 - x^k*A(x)^k)^k.

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A006195 Reversion of Jacobi theta_3.

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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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A184362 G.f.: eta(x) + x*eta'(x).

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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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A238440 Expansion of 1/E(q/E(q)) where E(q) = Product_{n>=1} (1 - q^n).

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A301577 G.f. A(x) satisfies: A(x) = Product_{k>=1} 1/(1 - kx^kA(x)^k).

A301625 G.f. A(x) satisfies: A(x) = Product_{k>=1} ((1 + x^kA(x)^k)/(1 - x^kA(x)^k))^k.