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.

A261647 Expansion of Product_{k>=0} ((1+x^(2*k+1))/(1-x^(2*k+1)))^3.

Original entry on oeis.org

1, 6, 18, 44, 102, 216, 428, 816, 1494, 2650, 4584, 7740, 12804, 20808, 33264, 52400, 81462, 125100, 189966, 285516, 425016, 627040, 917436, 1331856, 1919332, 2746926, 3905784, 5519352, 7754064, 10833192, 15055216, 20817600, 28647414, 39241336, 53517060
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 28 2015

Keywords

Links

Crossrefs

Cf. A015128, A156616.
Cf. A080054, A007096, A014969, A261648, A014970, A014972, A103261.
Cf. A261610, A261649, A261651.
Cf. A261611, A261650, A261652.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ exp(Pi*sqrt(3*n/2)) * 3^(1/4) / (8 * 2^(1/4) * n^(3/4)).

A261648 Expansion of Product_{k>=0} ((1+x^(2*k+1))/(1-x^(2*k+1)))^5.

Original entry on oeis.org

1, 10, 50, 180, 550, 1512, 3820, 9040, 20310, 43670, 90472, 181540, 354180, 674040, 1254640, 2289104, 4101430, 7228020, 12546030, 21473940, 36281656, 60565920, 99974140, 163297520, 264110180, 423211938, 672244600, 1059013320, 1655274320, 2568068120
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 28 2015

Keywords

Comments

In general, if j > 0 and g.f. = Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^j, then a(n) ~ exp(Pi*sqrt(j*n/2)) * j^(1/4) / (2^(j/2 + 7/4) * n^(3/4)).

Links

Crossrefs

Cf. A080054 (j=1), A007096 (j=2), A261647 (j=3), A014969 (j=4), A014970 (j=6), A014972 (j=8), A103261 (j=10).

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[((1+x^(2*k+1))/(1-x^(2*k+1)))^5,{k,0,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ exp(Pi*sqrt(5*n/2)) * 5^(1/4) / (16 * 2^(1/4) * n^(3/4)).

A261652 Expansion of Product_{k>=0} ((1+x^(4*k+1))/(1-x^(4*k+1)))^3.

Original entry on oeis.org

1, 6, 18, 38, 66, 108, 182, 306, 486, 728, 1068, 1578, 2318, 3312, 4614, 6388, 8862, 12192, 16488, 22038, 29400, 39156, 51702, 67554, 87810, 113982, 147384, 189200, 241446, 307356, 390408, 493662, 621006, 778712, 974628, 1216284, 1511756, 1872840, 2315538
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 28 2015

Keywords

Comments

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} ((1 + x^(a*k+b))/(1 - x^(a*k+b)))^j, then a(n) ~ Gamma(b/a)^j * 2^(j/2 - 3/2 - 2*b*j/a) * a^(-j/4 - 1/4 + b*j/(2*a)) * exp(Pi*sqrt(j*n/a)) * j^(1/4 - j/4 + b*j/(2*a)) * Pi^(b*j/a - j) * n^(j/4 - 3/4 - b*j/(2*a)).

Crossrefs

Cf. A015128 (a=1, b=1, j=1), A156616.
Cf. A080054 (a=2, b=1, j=1), A007096 (a=2, b=1, j=2), A261647 (a=2, b=1, j=3), A014969 (a=2, b=1, j=4), A261648 (a=2, b=1, j=5), A014970 (a=2, b=1, j=6), A014972 (a=2, b=1, j=8), A103261 (a=2, b=1, j=10).
Cf. A261610 (a=3, b=1, j=1), A261649 (a=3, b=1, j=2), A261651 (a=3, b=1, j=3).
Cf. A261611 (a=4, b=1, j=1), A261650 (a=4, b=1, j=2), A261652 (a=4, b=1, j=3).

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[((1+x^(4*k+1))/(1-x^(4*k+1)))^3,{k,0,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ exp(Pi*sqrt(3*n)/2) * 2^(1/4) * Gamma(1/4)^3 / (8 * 3^(1/8) * Pi^(9/4) * n^(3/8)).

A294592 a(n) = [x^n] (theta_3(x)/theta_4(x))^n, where theta_() is the Jacobi theta function.

Original entry on oeis.org

1, 4, 32, 304, 3072, 32024, 340352, 3666016, 39878656, 437091892, 4819567552, 53401892240, 594093969408, 6631726263608, 74242911364864, 833237193123104, 9371924860764160, 105614054423502408, 1192210691317862048, 13478559927485340144, 152589996020498655232, 1729590806617202662528
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 03 2017

Keywords

Links

Crossrefs

Cf. A007096, A014969, A014970, A014972, A066535, A080054, A103261, A270919, A289522, A291697.

Programs

  • Maple
    S:= series((JacobiTheta3(0,x)/JacobiTheta4(0,x))^n,x,51):
seq(coeff(S,x,n),n=0..50); # Robert Israel, Nov 03 2017
  • Mathematica
    Table[SeriesCoefficient[(EllipticTheta[3, 0, x]/EllipticTheta[4, 0, x])^n, {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[Product[((1 + x^(2 k + 1))/(1 - x^(2 k + 1)))^(2 n), {k, 0, n}], {x, 0, n}], {n, 0, 21}]
Table[SeriesCoefficient[(QPochhammer[-x, x^2]/QPochhammer[x, x^2])^(2 n), {x, 0, n}], {n, 0, 21}]
(* Calculation of constant d: *) 1/r /. FindRoot[{s == EllipticTheta[3, 0, r*s]/EllipticTheta[4, 0, r*s], EllipticTheta[4, 0, r*s] + r*s*Derivative[0, 0, 1][EllipticTheta][4, 0, r*s] == r*Derivative[0, 0, 1][EllipticTheta][3, 0, r*s]}, {r, 1/10}, {s, 2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Jan 17 2024 *)

Formula

a(n) = [x^n] Product_{k>=0} ((1 + x^(2*k+1))/(1 - x^(2*k+1)))^(2*n).
From Vaclav Kotesovec, Nov 05 2017: (Start)
a(n) ~ c * d^n / sqrt(n), where
d = 11.61255065799699699891360038489317237925475956178123836149123386457... and
c = 0.34456510029264878768512693687607064416428117641473856418257649837... (End)
Showing 1-4 of 4 results.

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