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.

Previous Showing 11-17 of 17 results.

A294395 E.g.f.: exp(Sum_{n>=1} A050999(n) * x^n).

Original entry on oeis.org

1, 1, 3, 67, 289, 5121, 71731, 861043, 18134817, 303946849, 6724342531, 146426154051, 3533373668353, 93259190078497, 2489644674735219, 75193364720030131, 2265438714279130561, 74716734198386887233, 2543592184722884351107, 90853513680763023292099
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2017

Keywords

Links

Crossrefs

E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): A294392 (k=0), A294394 (k=1), this sequence (k=2).
Cf. A050999, A262811.

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d^2*(d%2))*x^k))))

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A050999(k)*a(n-k)/(n-k)! for n > 0.
a(n) ~ (3*zeta(3))^(1/8) * n^(n - 1/8) / (2*exp(n - 4*zeta(3)^(1/4) * n^(3/4) / 3^(3/4) - n^(1/4) / (4*3^(5/4)*zeta(3)^(1/4)))). - Vaclav Kotesovec, Nov 01 2024

A294669 Expansion of Product_{k>=1} 1/(1 - x^(2*k-1))^(k*(3*k-1)/2).

Original entry on oeis.org

1, 1, 1, 6, 6, 18, 33, 55, 115, 185, 373, 604, 1113, 1903, 3251, 5678, 9350, 16153, 26420, 44561, 72912, 120150, 196329, 317988, 516881, 827778, 1333570, 2120492, 3381947, 5347513, 8447482, 13285450, 20813814, 32547272, 50638328, 78707858, 121738479
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 06 2017

Keywords

Links

Crossrefs

Cf. A035528, A262811, A294591, A278768.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(k*(3*k-1)/2),{k,1,nmax}],{x,0,nmax}],x]

Formula

a(n) ~ exp(2*Pi * n^(3/4) / (3*5^(1/4)) + Zeta(3) * sqrt(5*n) / Pi^2 + 5^(1/4) * (Pi/48 - 5*Zeta(3)^2 / Pi^5) * n^(1/4) + 100*Zeta(3)^3 / (3*Pi^8) + 17*Zeta(3) / (96*Pi^2) - 1/24) * sqrt(A) / (2^(101/48) * 5^(11/96) * Pi^(1/24) * n^(59/96)), where A is the Glaisher-Kinkelin constant A074962.

A285049 Expansion of Product_{k>=0} 1/(1-x^(5*k+1))^(5*k+1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 7, 7, 7, 7, 7, 18, 39, 39, 39, 39, 55, 121, 177, 177, 177, 198, 360, 591, 717, 717, 743, 1045, 1777, 2393, 2645, 2676, 3199, 4982, 7264, 8650, 9148, 9956, 13760, 20348, 26060, 28873, 30869, 38134, 54634, 73142, 85536, 92302, 106501, 143167
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Comments

In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1-x^(m*k-m+1))^(m*k-m+1), then a(n, m) ~ exp(c*m + 3 * 2^(-2/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(19*m/36 + 1/(6*m) - 1) * m^(17*m/36 + 5/(6*m) - 3/2) * Pi^(m/2 - 1) * Zeta(3)^(1/(6*m) + m/36) / (sqrt(3) * Gamma(1/m)^(m-1) * n^(1/2 + 1/(6*m) + m/36)), where c = Integral_{x=0..infinity} exp((2*m-1)*x) / (x*(exp(m*x) - 1)^2) + (1/12 - (m-1)^2/(2*m^2))/(x*exp(x)) - 1/(m^2*x^3) - (m-1)/(m^2*x^2) dx. - Vaclav Kotesovec, Apr 17 2017

Links

Crossrefs

Product_{k>=0} 1/(1-x^(m*k+1))^(m*k+1): A000219 (m=1), A262811 (m=2), A262947 (m=3), A285048 (m=4), this sequence (m=5).
Cf. A285071.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1-x^(5*k-4))^(5*k-4), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 16 2017 *)

Formula

a(n) ~ 2^(301/180) * 5^(37/36) * Pi^(3/2) * Zeta(3)^(31/180) * exp(5*c + 3 * 2^(-2/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) / (sqrt(3) * Gamma(1/5)^4 * n^(121/180)), where c = Integral_{x=0..inf} ((-71/(exp(x)*300) + 1/(exp(x)*(1 - exp(-5*x))^2) - 1/(25*x^2) - 4/(25*x))/x) dx = 0.186382690624752630391368364629918483384424086341764409146923686... - Vaclav Kotesovec, Apr 16 2017

A285132 Expansion of Product_{k>=0} 1/(1-x^(5*k+4))^(5*k+4).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 0, 10, 9, 0, 0, 20, 36, 14, 0, 35, 90, 101, 19, 56, 180, 320, 202, 108, 315, 730, 859, 492, 533, 1390, 2300, 2139, 1354, 2393, 4835, 6475, 5098, 4619, 8813, 14926, 16395, 12982, 15751, 28962, 41162, 40256, 35200, 51731, 85365, 106145
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Comments

In general, if m > 1 and g.f. = Product_{k>=1} 1/(1-x^(m*k-1))^(m*k-1), then a(n, m) ~ exp(c*m + 3 * 2^(-2/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * (2*Zeta(3))^(1/(6*m) + m/36) / (sqrt(3) * Gamma(1 - 1/m) * m^(1/2 - 5/(6*m) + m/36) * n^(1/2 + 1/(6*m) + m/36)), where c = Integral_{x=0..infinity} exp((m+1)*x) / (x*(exp(m*x)-1)^2) + (1/12 - 1/(2*m^2))/(x*exp(x)) - 1/(m^2*x^3) - 1/(m^2*x^2) dx. - Vaclav Kotesovec, Apr 17 2017

Links

Crossrefs

Product_{k>=0} 1/(1-x^(m*k+m-1))^(m*k+m-1): A262811 (m=2), A262946 (m=3), A285131 (m=4), this sequence (m=5).
Cf. A285214.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1/(1-x^(5*k-1))^(5*k-1), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
  • PARI
    x='x+O('x^100); Vec(prod(k=0, 100, 1/(1 - x^(5*k + 4))^(5*k + 4))) \\ Indranil Ghosh, Apr 15 2017

Formula

a(n) ~ exp(5*c + 3*2^(-2/3)*5^(-1/3)*Zeta(3)^(1/3)*n^(2/3)) * (2*Zeta(3))^(31/180) / (sqrt(3) * 5^(17/36) * Gamma(4/5) * n^(121/180)), where c = Integral_{x=0..inf} ((19/(exp(x)*300) + 1/(exp(4*x)*(1-exp(-5*x))^2) - 1/(25*x^2) - 1/(25*x))/x) dx = -0.12699586713882325294527057473113580561183418857868946729897216431919... - Vaclav Kotesovec, Apr 15 2017

A285246 Expansion of Product_{k>=1} (1 - x^(5*k))^(5*k) / (1 - x^k)^k.

Original entry on oeis.org

1, 1, 3, 6, 13, 19, 43, 71, 130, 217, 380, 619, 1049, 1685, 2757, 4404, 7027, 11014, 17326, 26820, 41488, 63514, 96970, 146808, 221659, 332212, 496439, 737535, 1091938, 1608564, 2361929, 3452736, 5031138, 7302373, 10566038, 15234196, 21900182, 31380435
Offset: 0

Views

Author

Seiichi Manyama, Apr 15 2017

Keywords

Comments

In general, if m > 1 and g.f. = Product_{k>=1} (1 - x^(m*k))^(m*k)/((1 - x^k)^k), then a(n, m) ~ exp(1/12 - m/12 + 3 * 2^(-2/3) * (1-1/m)^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(-(m+11)/36) * A^(m-1) * (m-1)^((7-m)/36) * m^(-(2*m+7)/36) * Zeta(3)^((7-m)/36) * n^((m-25)/36) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 16 2017

Links

Crossrefs

Product_{k>=1} (1 - x^(m*k))^(m*k)/(1 - x^k)^k: A262811 (m=2), A262923 (m=3), A285215 (m=4), this sequence (m=5).
Cf. A285263, A285285.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1 / ((1-x^(5*k+1))^(5*k+1) * (1-x^(5*k+2))^(5*k+2) * (1-x^(5*k+3))^(5*k+3) * (1-x^(5*k+4))^(5*k+4)), {k,0,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(5*k))^(5*k)/((1 - x^k)^k), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *)
  • PARI
    x='x+O('x^100); Vec(prod(k=0, 100, 1 / ((1 - x^(5*k + 1))^(5*k + 1)*(1 - x^(5*k + 2))^(5*k + 2)*(1 - x^(5*k + 3))^(5*k + 3)*(1 - x^(5*k + 4))^(5*k + 4)))) \\ Indranil Ghosh, Apr 15 2017

Formula

G.f.: Product_{k>=0} 1 / ((1-x^(5*k+1))^(5*k+1) * (1-x^(5*k+2))^(5*k+2) * (1-x^(5*k+3))^(5*k+3) * (1-x^(5*k+4))^(5*k+4)).
a(n) ~ exp(-1/3 + 3*(Zeta(3)/5)^(1/3)*n^(2/3)) * A^4 * Zeta(3)^(1/18) / (2^(1/3) * 5^(17/36) * sqrt(3*Pi) * n^(5/9)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 16 2017

A294668 Expansion of Product_{k>=1} 1/(1 - x^(2*k-1))^(k*(3*k+1)/2).

Original entry on oeis.org

1, 2, 3, 11, 19, 42, 93, 170, 352, 658, 1266, 2351, 4316, 7926, 14146, 25458, 44748, 78687, 136747, 235988, 405139, 689108, 1168260, 1963940, 3289950, 5474700, 9070976, 14954802, 24537752, 40099905, 65225553, 105713691, 170600344, 274367688, 439568770, 701867457
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 06 2017

Keywords

Links

Crossrefs

Cf. A035528, A262811, A294591, A294667, A294669.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1 - x^(2*k-1))^(k*(3*k+1)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(2*Pi * n^(3/4) / (3*5^(1/4)) + 2*Zeta(3) * sqrt(5*n) / Pi^2 + 5^(1/4)*(5*Pi/48 - 20*Zeta(3)^2 / Pi^5) * n^(1/4) + 800 * Zeta(3)^3 / (3*Pi^8) - 73*Zeta(3) / (96*Pi^2) - 1/12) * A / (2^(115/48) * 5^(5/48) * Pi^(1/12) * n^(29/48)), where A is the Glaisher-Kinkelin constant A074962.

A318968 Expansion of exp(Sum_{k>=1} ( Sum_{d|k, d odd} d^k ) * x^k/k).

Original entry on oeis.org

1, 1, 1, 10, 10, 635, 797, 118446, 124071, 43174194, 45404910, 25982930761, 26443958420, 23324558686914, 23640266984002, 29216576615057082, 29447535265299613, 48690644491136860817, 48980258924147884960, 104176334607664412086539, 104636388540330684649083, 278323070872780066332365486
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 06 2018

Keywords

Crossrefs

Cf. A000009, A023881, A206303, A262811, A292919, A318969.

Programs

  • Mathematica
    nmax = 21; CoefficientList[Series[Exp[Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 21; CoefficientList[Series[Product[1/(1 - (2 k - 1)^(2 k - 1) x^(2 k - 1))^(1/(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[Mod[d, 2] d^k, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 21}]

Formula

G.f.: Product_{k>=1} 1/(1 - (2*k - 1)^(2*k-1)*x^(2*k-1))^(1/(2*k-1)).
Previous Showing 11-17 of 17 results.

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