A066398 -id:A066398 - cp's OEIS frontend

A007312 Reversion of g.f. (with constant term omitted) for partition numbers.

1, -2, 5, -15, 52, -200, 825, -3565, 15900, -72532, 336539, -1582593, 7524705, -36111810, 174695712, -851020367, 4171156249, -20555470155, 101787990805, -506227992092, 2527493643612, -12663916942984, 63656297034920, -320914409885850, 1622205233276889

Offset: 1

N. J. A. Sloane, Mira Bernstein

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vaclav Kotesovec, Table of n, a(n) for n = 1..1000
Index entries for reversions of series

Cf. A000041, A050393, A066398, A334315.

# Using function CompInv from A357588.
CompInv(25, n -> combinat:-numbpart(n)); # Peter Luschny, Oct 05 2022

nmax = 30; Rest[CoefficientList[InverseSeries[Series[Sum[PartitionsP[n]*x^n, {n, 1, nmax}], {x, 0, nmax}]], x]] (* Vaclav Kotesovec, Nov 11 2017 *)
Rest[CoefficientList[InverseSeries[Series[-1 + 1/QPochhammer[x],{x,0,30}],x],x]] (* Vaclav Kotesovec, Jan 18 2024 *)
(* Calculation of constant d: *) Chop[1/r /. FindRoot[{(1 + r)*QPochhammer[s, s] == 1, Log[1 - s] + QPolyGamma[0, 1, s] - (1 + r)*s*Log[s] * Derivative[0, 1][QPochhammer][s, s] == 0}, {r, -1/5}, {s, -1/2}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 18 2024 *)

From Vaclav Kotesovec, Nov 11 2017: (Start)
a(n) ~ -(-1)^n * c * d^n / n^(3/2), where
d = 5.379264118840884783404842050140885100801253519243086... and
c = 0.10697042824132534557642152089737206588353695053... (End)
G.f. A(x) satisfies: A(x) = 1 - (1/(1 + x)) * Product_{k>=2} 1/(1 - A(x)^k). - Ilya Gutkovskiy, Apr 23 2020

Signs corrected Dec 24 2001

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

A184365 G.f.: eta(x) - x*eta'(x), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

Original entry on oeis.org

1, 0, 1, 0, 0, -4, 0, -6, 0, 0, 0, 0, 11, 0, 0, 14, 0, 0, 0, 0, 0, 0, -21, 0, 0, 0, -25, 0, 0, 0, 0, 0, 0, 0, 0, 34, 0, 0, 0, 0, 39, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -50, 0, 0, 0, 0, 0, -56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 69, 0, 0, 0, 0, 0, 0, 76, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -91, 0, 0, 0

Offset: 0

Paul D. Hanna, Jan 17 2011

sign

			G.f.: A(x) = 1 + x^2 - 4*x^5 - 6*x^7 + 11*x^12 + 14*x^15 - 21*x^22 - 25*x^26 + 34*x^35 + 39*x^40 - 50*x^51 +...
Illustrate the property: [x^(n+1)] A(x)*eta(x)^n = 0
in the table of coefficients of A(x)*eta(x)^n for n=0..10:
[1,(0), 1, 0, 0, -4, 0, -6, 0, 0, 0, 0, 11, 0, 0, 14,...];
[1, -1,(0), -1, -1, -3, 4, 0, 6, 7, -4, 0, 0, -11,...];
[1, -2, 0,(0), 0, 0, 7, 0, 0, 0, -21, 0, 0, 0, 0, 44,...];
[1, -3, 1, 2,(0), 1, 5, -6, -9, 0, -21, 28, 20, 9,...];
[1, -4, 3, 4, -3,(0), 1, -10, -9, 16, -9, 54, 7, -40,...];
[1, -5, 6, 5, -10, 0,(0), -7, 0, 35, -12, 45, -49, -105,...];
[1, -6, 10, 4, -21, 6, 5,(0), 7, 38, -42, 12, -90, -96,...];
[1, -7, 15, 0, -35, 24, 14,0,(0), 20, -77, 0, -55, 0,...];
[1, -8, 21, -8, -50, 60, 18,-22,-21,(0), -73, 36, 45, 76,...];
[1, -9, 28, -21, -63, 119, 0, -78,-33,14,(0), 77, 119, 0,...];
[1, -10, 36, -40, -70, 204, -65, -168,15,90,117,(0),...]; ...
so that the coefficient of x^(n+1) in A(x)*eta(x)^n is zero for n>=0.
Note: the g.f.s of the diagonals in the above table are powers of G(x), where G(x) = eta(x*G(x)) is the g.f. of A066398.
The g.f. of A184366 equals:
A(x)*eta(x)^2 = 1 - 2*x + 7*x^6 - 21*x^10 + 44*x^15 - 78*x^21 + 125*x^28 - 187*x^36 +...+ -(-1)^n*(n-2)(n+3)(2n+1)/6*x^(n(n+1)/2) +...

Cf. A184366, A184362, A066398.

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

G.f. satisfies: [x^(n+1)] A(x)*eta(x)^n = 0 for n>=0.
G.f.: A(x) = 1 - Sum_{n>=1} (-1)^n*((n-1)*(3*n+2) + (n+1)*(3*n-2)*x^n)/2 * x^(n*(3*n-1)/2).
G.f.: A(x) = G(x)/eta(x)^2 where G(x) = Sum_{n>=0} -(-1)^n*(n-2)(n+3)(2n+1)/6*x^(n(n+1)/2) is the g.f. of A184366.

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.

A334315 E.g.f. A(x) satisfies: A(x) = x - Sum_{k>=2} p(k) * A(x)^k / k!, where p = A000041 (partition numbers).

Original entry on oeis.org

1, -2, 9, -65, 653, -8432, 133188, -2488450, 53683569, -1313214351, 35916970957, -1086055854233, 35975402985863, -1295514629022924, 50391598721116365, -2105485003413499952, 94047072252968125326, -4472183077495496587696, 225565085807090517308839

Offset: 1

Ilya Gutkovskiy, Apr 22 2020

sign

Exponential reversion of A000041 (partition numbers).

Index entries for reversions of series

Cf. A000041, A007312, A050394, A066398.

nmax = 19; CoefficientList[InverseSeries[Series[Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}], {x, 0, nmax}], x], x] Range[0, nmax]! // Rest

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A007312 Reversion of g.f. (with constant term omitted) for partition numbers.

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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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A184365 G.f.: eta(x) - x*eta'(x), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

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

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A334315 E.g.f. A(x) satisfies: A(x) = x - Sum_{k>=2} p(k) * A(x)^k / k!, where p = A000041 (partition numbers).

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