A171802 -id:A171802 - cp's OEIS frontend

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).

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)

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

Original entry on oeis.org

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

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)

A366026 G.f. A(x) satisfies A(x) = Product_{k>=1} (1 + x^kA(x)^(2k)).

Original entry on oeis.org

1, 1, 3, 13, 64, 340, 1903, 11053, 65992, 402508, 2497207, 15709873, 99980007, 642535004, 4164018953, 27181480712, 178559253274, 1179546465168, 7830695860690, 52216823047741, 349584244515573, 2348869478981267, 15833924106623011, 107057382854642578, 725829177205070854

Offset: 0

Vaclav Kotesovec, Sep 26 2023

nonn

Cf. A171802, A181315.

nmax = 30; A[] = 0; Do[A[x] = Product[1 + x^k*A[x]^(2*k), {k, 1, nmax}] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
(* The constants {d,c}: *) {1/r, 1/(2*Sqrt[Pi*(1/s^2 + 2*r^2*s*Derivative[0, 2][QPochhammer][-1, r*s^2])])} /. FindRoot[{2*s == QPochhammer[-1, r*s^2], r*s*Derivative[0, 1][QPochhammer][-1, r*s^2] == 1}, {r, 1/8}, {s, 1}, WorkingPrecision -> 120]

A(x) satisfies QPochhammer(-1, x*A(x)^2) = 2*A(x).

a(n) ~ c * d^n / n^(3/2), where d = 7.2188305975020061051473056449576894316519... and c = 0.2182691546096422371919544994005940622002...

cp's OEIS Frontend

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).

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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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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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A366026 G.f. A(x) satisfies A(x) = Product_{k>=1} (1 + x^k*A(x)^(2*k)).

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A366026 G.f. A(x) satisfies A(x) = Product_{k>=1} (1 + x^kA(x)^(2k)).