cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Views

Author

Vladimir Reshetnikov, Nov 21 2016

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

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

Views

Author

Paul D. Hanna, Dec 19 2009

Keywords

Examples

			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 +...

Links

Crossrefs

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

Programs

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

Formula

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

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

Views

Author

Paul D. Hanna, Dec 19 2009

Keywords

Examples

			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 +...

Crossrefs

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

Programs

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

Formula

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)

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

Views

Author

Paul D. Hanna, Dec 20 2009

Keywords

Examples

			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)

Crossrefs

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

Programs

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

Formula

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)
Showing 1-4 of 4 results.

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