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.

A294500 Binomial transform of the number of planar partitions (A000219).

Original entry on oeis.org

1, 2, 6, 19, 60, 185, 559, 1662, 4875, 14134, 40564, 115370, 325465, 911355, 2534595, 7004827, 19246626, 52596377, 143006632, 386984573, 1042537831, 2796803110, 7473161196, 19893461042, 52767059608, 139488323734, 367540167625, 965445514862, 2528516552660
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2017

Keywords

Comments

Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
Special cases:
p < 1/2, g(n) = 0
p = 1/2, g(n) = r^2/16
p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536
p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625

Links

Crossrefs

Cf. A218481, A266232, A294501, A294502, A294504.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A000219(k).
a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) - Zeta(3)/12) * 2^(n + 7/18) * Zeta(3)^(7/36) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.
G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)). - Ilya Gutkovskiy, Aug 20 2018

A294503 Inverse binomial transform of A026007.

Original entry on oeis.org

1, 0, 1, 1, -3, 10, -23, 48, -92, 171, -321, 626, -1265, 2576, -5099, 9478, -15925, 22617, -21816, -8506, 121659, -436121, 1204710, -2962759, 6860591, -15427559, 34323613, -76269455, 169591278, -376162414, 827819644, -1798045927, 3839392935, -8041078328
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2017

Keywords

Links

Crossrefs

Cf. A026007, A294501, A294502, A294505.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[(1+x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[(-1)^(n-k) * Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A026007(k).
G.f.: (1/(1 + x))*Product_{k>=1} (1 + x^k/(1 + x)^k)^k. - Ilya Gutkovskiy, Aug 20 2018

A294504 Binomial transform of A156616.

Original entry on oeis.org

1, 3, 11, 41, 147, 509, 1717, 5671, 18395, 58735, 184961, 575337, 1769981, 5390997, 16270587, 48696299, 144620059, 426428645, 1249007767, 3635595953, 10520770265, 30278391475, 86689798089, 246988386691, 700439171501, 1977660342139, 5560497703461
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2017

Keywords

Links

Crossrefs

Cf. A156616, A218481, A294500, A294502, A294505.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A156616(k).
a(n) ~ exp(3 * (7*Zeta(3))^(1/3) * n^(2/3) / 4 + (7*Zeta(3))^(2/3) * n^(1/3) / 8 + 1/12 - 7*Zeta(3)/48) * (7*Zeta(3))^(7/36) * 2^(n - 1/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.
G.f.: (1/(1 - x))*exp(Sum_{k>=1} (sigma_2(2*k) - sigma_2(k))*x^k/(2*k*(1 - x)^k)). - Ilya Gutkovskiy, Oct 15 2018

A320564 Expansion of Product_{k>=1} (1 + x^k/(1 - x)^k)^k.

Original entry on oeis.org

1, 1, 3, 10, 30, 87, 249, 705, 1974, 5471, 15032, 40997, 111079, 299151, 801139, 2134251, 5657895, 14930596, 39232009, 102673794, 267692321, 695440442, 1800582809, 4646964755, 11956293758, 30673060344, 78470890246, 200218512582, 509557661691, 1293664233400, 3276659862518
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

Comments

First differences of the binomial transform of A026007.

Links

Crossrefs

Cf. A026007, A129519, A294502, A320563.

Programs

  • Maple
    seq(coeff(series(mul((1+x^k/(1-x)^k)^k,k=1..n),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 15 2018
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[(1 + x^k/(1 - x)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) (1 - x)^k x^k/(k ((1 - x)^k - x^k)^2), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*(1 - x)^k*x^k/(k*((1 - x)^k - x^k)^2)).
a(n) ~ Zeta(3)^(1/6) * 2^(n - 13/12) * exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3)/4 + (3*Zeta(3))^(2/3) * n^(1/3)/8 - Zeta(3)/16) / (3^(1/3) * sqrt(Pi) * n^(2/3)). - Vaclav Kotesovec, Oct 15 2018

A307259 Expansion of (1/(1 - x)) * Product_{k>=1} (1 + k*x^k/(1 - x)^k).

Original entry on oeis.org

1, 2, 5, 15, 44, 126, 357, 1003, 2783, 7618, 20627, 55421, 148021, 393140, 1038123, 2724992, 7112022, 18465708, 47726767, 122861732, 315123476, 805428727, 2051556778, 5207982062, 13177117709, 33235023381, 83574705456, 209576713721, 524181331710, 1307849984089, 3255539133109
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 01 2019

Keywords

Comments

Binomial transform of A022629.

Crossrefs

Cf. A022629, A266232, A294502, A307260, A318127.

Programs

  • Maple
    a:=series((1/(1-x))*mul(1+k*x^k/(1-x)^k,k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 03 2019
  • Mathematica
    nmax = 30; CoefficientList[Series[1/(1 - x) Product[(1 + k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) = Sum_{k=0..n} binomial(n,k)*A022629(k).
Showing 1-5 of 5 results.

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