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 1-5 of 5 results.

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)

A304443 Coefficient of x^n in Product_{k>=1} (1+x^k)^(2*n).

Original entry on oeis.org

1, 2, 10, 62, 394, 2562, 16966, 113794, 770442, 5254334, 36042250, 248403586, 1718732998, 11931569028, 83064794746, 579696375972, 4054279504266, 28408328186508, 199390547044342, 1401564307833908, 9865190079554954, 69522550703432476, 490484539061916794
Offset: 0

Views

Author

Vaclav Kotesovec, May 12 2018

Keywords

Links

Crossrefs

Cf. A022567, A026011, A270913, A304444.

Programs

  • Mathematica
    nmax = 25; Table[SeriesCoefficient[Product[(1+x^k)^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 25; Table[SeriesCoefficient[(QPochhammer[-1, x]/2)^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d,c}: *) {1/r, Sqrt[Derivative[0, 1][QPochhammer][-1, r*s] / (Pi*r*(Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s]^2 + 2*s*Derivative[0, 2][QPochhammer][-1, r*s]))]} /. FindRoot[{4*s == QPochhammer[-1, r*s]^2, 2*r*Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s] == 2}, {r, 1/8}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

Formula

a(n) ~ c * d^n / sqrt(n), where d = 7.21883059750200610514730564495768943165197819880185778427663522275469... and c = 0.300860732623379969554285615234449502629772950943717460278989499...

A327215 Self-convolution of A008485.

Original entry on oeis.org

1, 2, 11, 54, 279, 1442, 7530, 39474, 207693, 1095522, 5790116, 30650038, 162451560, 861920492, 4577055823, 24323292984, 129338944225, 688128700422, 3662798123481, 19504467792378, 103899170100154, 553642311668244, 2951010332435759, 15733439067954134
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 26 2019

Keywords

Links

Crossrefs

Cf. A000712, A008485, A304444, A309986, A327214.

Programs

  • Mathematica
    A008485[n_]:=SeriesCoefficient[Product[1/(1-x^k)^n, {k, 1, n}], {x, 0, n}];
Table[Sum[A008485[k]*A008485[n-k], {k, 0, n}], {n, 0, 25}]

Formula

a(n) ~ c^2 * Pi * d^n, where d = A270915 = 5.352701333486642687772415814165... and c = A327279 = 0.26801521271073331568695383828... (see A008485).

A304447 Coefficient of x^n in Product_{k>=1} ((1+x^k)/(1-x^k))^(2*n).

Original entry on oeis.org

1, 4, 40, 448, 5264, 63624, 783328, 9770240, 123040288, 1561033348, 19922193200, 255472920256, 3289122824000, 42488488508808, 550435283089088, 7148519205631488, 93038785849116736, 1213215382135324680, 15846906866928513736, 207302985358274247104
Offset: 0

Views

Author

Vaclav Kotesovec, May 12 2018

Keywords

Links

Crossrefs

Cf. A261386, A270919, A304443, A304444, A304448.

Programs

  • Mathematica
    nmax = 20; Table[SeriesCoefficient[Product[((1+x^k)/(1-x^k))^(2*n), {k, 1, n}], {x, 0, n}], {n, 0, nmax}]
nmax = 20; Table[SeriesCoefficient[(QPochhammer[-1, x]/2/QPochhammer[x])^(2*n), {x, 0, n}], {n, 0, nmax}]
(* Calculation of constants {d,c}: *) eq = FindRoot[{QPochhammer[-1, r*s] == 2*Sqrt[s]*QPochhammer[r*s], (QPochhammer[ r*s]*(Log[r*s] - 2*Log[1 - r*s] - 2*QPolyGamma[0, 1, r*s])) / Log[r*s] - r*Sqrt[s]*Derivative[0, 1][QPochhammer][-1, r*s] + 2*r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == 0}, {r, 1/12}, {s, 2}, WorkingPrecision -> 1000]; {N[1/r /. eq, 120], val = Sqrt[((1 - r*s)*Log[r*s]^2*QPochhammer[r*s]) / (Pi*(2*r*s*(-1 + r*s) * Log[r*s]*(2*(Log[r*s] - 2*Log[1 - r*s] - 2*QPolyGamma[0, 1, r*s]) * Derivative[0, 1][QPochhammer][r*s, r*s] + r*Sqrt[s]*Log[r*s] * (-Derivative[0, 2][QPochhammer][-1, r*s] + 2*Sqrt[s]*Derivative[0, 2][QPochhammer][r*s, r*s])) + QPochhammer[ r*s]*(16*r*s*ArcTanh[1 - 2*r*s] + (1 - r*s)*Log[r*s]^2 - 8*Log[1 - r*s] + 4*(-1 + r*s)*Log[1 - r*s]^2 + 8*(-1 + r*s)*(1 + Log[1 - r*s])* QPolyGamma[0, 1, r*s] + 4*(-1 + r*s)*QPolyGamma[0, 1, r*s]^2 + 4*(-1 + r*s)*(QPolyGamma[1, 1, r*s] - 2*r*s*Log[r*s]*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))))] /. eq; N[Chop[val], -Floor[Log[10, Abs[Im[val]]]] - 3]} (* Vaclav Kotesovec, Oct 03 2023 *)

Formula

a(n) ~ c * d^n / sqrt(n), where d = 13.43567525239504624062504283058713960962824709850658926621911428148173077464... and c = 0.3323527904383991069791889982282236666403568774227549868882810268779...
Showing 1-5 of 5 results.

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