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 1-9 of 9 results.

A218481 Binomial transform of the partition numbers (A000041).

Original entry on oeis.org

1, 2, 5, 13, 34, 88, 225, 569, 1425, 3538, 8717, 21331, 51879, 125474, 301929, 723144, 1724532, 4096210, 9693455, 22859524, 53733252, 125919189, 294232580, 685661202, 1593719407, 3695348909, 8548564856, 19732115915, 45450793102, 104481137953, 239718272765
Offset: 0

Views

Author

Paul D. Hanna, Oct 29 2012

Keywords

Comments

Partial sums of A218482.
From Vaclav Kotesovec, Nov 02 2023: (Start)
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
(End)

Examples

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 13*x^3 + 34*x^4 + 88*x^5 + 225*x^6 + 569*x^7 +...
The g.f. equals the product:
A(x) = 1/((1-x)-x) * (1-x)^2/((1-x)^2-x^2) * (1-x)^3/((1-x)^3-x^3) * (1-x)^4/((1-x)^4-x^4) * (1-x)^5/((1-x)^5-x^5) * (1-x)^6/((1-x)^6-x^6) * (1-x)^7/((1-x)^7-x^7) *...
and also equals the series:
A(x) = 1/(1-x) * (1  +  x*(1-x)/((1-x)-x)^2  +  x^4*(1-x)^2/(((1-x)-x)*((1-x)^2-x^2))^2  +  x^9*(1-x)^3/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3))^2  +  x^16*(1-x)^4/(((1-x)-x)*((1-x)^2-x^2)*((1-x)^3-x^3)*((1-x)^4-x^4))^2 +...).
The terms begin:
a(0) = 1*1,
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 2*1 + 1*2 = 5;
a(3) = 1*1 + 3*1 + 3*2 + 1*3 = 13;
a(4) = 1*1 + 4*1 + 6*2 + 4*3 + 1*5 = 34; ...

Links

Crossrefs

Cf. A218482, A222115, A000041, A000203, A266497.
Cf. A294466, A281425, A095051.
Cf. A266232, A294467, A293467, A294468, A294500.

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]*PartitionsP[k],{k,0,n}],{n,0,30}] (* Vaclav Kotesovec, Jun 25 2015 *)
nmax = 30; CoefficientList[Series[Sum[PartitionsP[k] * x^k / (1-x)^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2022 *)
  • PARI
    {a(n)=sum(k=0,n,binomial(n,k)*numbpart(k))}
for(n=0,40,print1(a(n),", "))
  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*prod(k=1,n,(1-x)^k/((1-x)^k-X^k)),n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*sum(m=0,n,x^m*(1-x)^(m*(m-1)/2)/prod(k=1,m,((1-x)^k - X^k))),n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*sum(m=0,n,x^(m^2)*(1-X)^m/prod(k=1,m,((1-x)^k - x^k)^2)),n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*exp(sum(m=1,n+1,x^m/((1-x)^m-X^m)/m)),n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*exp(sum(m=1,n+1,sigma(m)*x^m/(1-X)^m/m)),n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n));polcoeff(1/(1-X)*prod(k=1,n,(1 + x^k/(1-X)^k)^valuation(2*k,2)),n)}

Formula

G.f.: 1/(1-x)*Product_{n>=1} (1-x)^n / ((1-x)^n - x^n).
G.f.: 1/(1-x)*Sum_{n>=0} x^n * (1-x)^(n*(n-1)/2) / Product_{k=1..n} ((1-x)^k - x^k).
G.f.: 1/(1-x)*Sum_{n>=0} x^(n^2) * (1-x)^n / Product_{k=1..n} ((1-x)^k - x^k)^2.
G.f.: 1/(1-x)*exp( Sum_{n>=1} x^n/((1-x)^n - x^n) / n ).
G.f.: 1/(1-x)*exp( Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n ), where sigma(n) is the sum of divisors of n (A000203).
G.f.: 1/(1-x)*Product_{n>=1} (1 + x^n/(1-x)^n)^A001511(n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
Logarithmic derivative yields A222115.
a(n) ~ exp(Pi*sqrt(n/3) + Pi^2/24) * 2^(n-1) / (n*sqrt(3)). - Vaclav Kotesovec, Jun 25 2015

A266232 Binomial transform of the number of partitions into distinct parts (A000009).

Original entry on oeis.org

1, 2, 4, 9, 21, 49, 114, 265, 615, 1422, 3272, 7493, 17090, 38850, 88065, 199097, 448953, 1009788, 2265642, 5071611, 11328395, 25254093, 56195143, 124829822, 276839061, 612991848, 1355268779, 2992016128, 6596222234, 14522634554, 31933047707, 70130243427
Offset: 0

Views

Author

Vaclav Kotesovec, Dec 25 2015

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. A000009, A294467, A293467, A294468.
Cf. A218481, A294466, A281425, A095051, A294500.

Programs

  • Mathematica
    Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}]
nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] * x^k / (1-x)^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2022 *)

Formula

a(n) ~ 2^(n-5/4) * exp(Pi*sqrt(n/6) + Pi^2/48) / (3^(1/4)*n^(3/4)).
G.f.: (1/(1 - x))*Product_{k>=1} (1 + x^k/(1 - x)^k). - Ilya Gutkovskiy, Aug 19 2018

A294502 Binomial transform of A026007.

Original entry on oeis.org

1, 2, 5, 15, 45, 132, 381, 1086, 3060, 8531, 23563, 64560, 175639, 474790, 1275929, 3410180, 9068075, 23998671, 63230680, 165904474, 433596795, 1129037237, 2929620046, 7576584801, 19532878559, 50205938903, 128676829149, 328895341731, 838453003422
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2017

Keywords

Links

Crossrefs

Cf. A026007, A218481, A294500, A294503, A294504.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[(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) * A026007(k).
a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 4 + (3*Zeta(3))^(2/3) * n^(1/3) / 8 - Zeta(3)/16) * Zeta(3)^(1/6) * 2^(n - 1/12) / (3^(1/3) * sqrt(Pi) * n^(2/3)).
G.f.: (1/(1 - x))*Product_{k>=1} (1 + x^k/(1 - x)^k)^k. - Ilya Gutkovskiy, Aug 19 2018

A294501 Inverse binomial transform of the number of planar partitions (A000219).

Original entry on oeis.org

1, 0, 2, -1, 4, -7, 19, -48, 123, -304, 728, -1694, 3865, -8735, 19739, -44875, 102818, -236939, 546988, -1260023, 2888607, -6584008, 14927816, -33714166, 75976024, -171095098, 385405617, -868708176, 1959010348, -4417777937, 9957188242, -22420045445
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2017

Keywords

Links

Crossrefs

Cf. A281425, A293467, A294500, A294503, A294505.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[1/(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) * A000219(k).
G.f.: (1/(1 + x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 + x)^k)). - Ilya Gutkovskiy, Aug 20 2018

A318127 Expansion of (1/(1 - x)) * Product_{k>=1} 1/(1 - k*x^k/(1 - x)^k).

Original entry on oeis.org

1, 2, 6, 19, 61, 191, 588, 1785, 5351, 15868, 46628, 135921, 393318, 1130538, 3229753, 9175347, 25931605, 72936434, 204223348, 569427145, 1581458917, 4375905243, 12065914843, 33160240020, 90848002909, 248154744196, 675932128695, 1836182233332, 4975249827916, 13447775233746
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 18 2018

Keywords

Comments

Binomial transform of A006906.

Links

Crossrefs

Cf. A006906, A218481, A294500.

Programs

  • Maple
    a:=series(1/(1-x)*mul(1/(1-k*x^k/(1-x)^k),k=1..100),x=0,30): seq(coeff(a,x,n),n=0..29); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 29; CoefficientList[Series[1/(1 - x) Product[1/(1 - k x^k/(1 - x)^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[1/(1 - x) Exp[Sum[Sum[j^k x^(k j)/(k (1 - x)^(k j)), {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[Binomial[n, k] Total[Times @@@ IntegerPartitions[k]], {k, 0, n}], {n, 0, 29}]

Formula

G.f.: (1/(1 - x))*exp(Sum_{k>=1} Sum_{j>=1} j^k*x^(k*j)/(k*(1 - x)^(k*j))).
a(n) = Sum_{k=0..n} binomial(n,k)*A006906(k).
a(n) ~ c * (1 + 3^(1/3))^n, where c = 97923.037496367052161042295948902147352859984491653037730624387144966464... = 1/((3^(1/3) - 1) * (3^(2/3) - 2)) * Product_{k>=4} 1/(1 - k/3^(k/3)). - Vaclav Kotesovec, Aug 19 2018

A320563 Expansion of Product_{k>=1} 1/(1 - x^k/(1 - x)^k)^k.

Original entry on oeis.org

1, 1, 4, 13, 41, 125, 374, 1103, 3213, 9259, 26430, 74806, 210095, 585890, 1623240, 4470232, 12241799, 33349751, 90410255, 243977941, 655553258, 1754265279, 4676358086, 12420299846, 32873598566, 86721264126, 228051843891, 597905347237, 1563071037798, 4074973824099
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

Comments

First differences of the binomial transform of A000219.

Links

Crossrefs

Cf. A000219, A001157, A103446, A218482, A294500, A320564.

Programs

  • Maple
    seq(coeff(series(mul((1-x^k/(1-x)^k)^(-k),k=1..n),x,n+1), x, n), n = 0 .. 29); # Muniru A Asiru, Oct 15 2018
  • Mathematica
    nmax = 29; CoefficientList[Series[Product[1/(1 - x^k/(1 - x)^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k] x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)).
a(n) ~ Zeta(3)^(7/36) * 2^(n - 11/18) * exp(3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) + (1 - Zeta(3))/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Oct 15 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

A294529 Binomial transform of A001156.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 86, 192, 420, 905, 1939, 4163, 8987, 19494, 42368, 91990, 199127, 429345, 921982, 1972553, 4206909, 8949412, 19001874, 40293048, 85373962, 180826115, 382957231, 811027414, 1717497958, 3636335170, 7695599294, 16275268520, 34389570596
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 02 2017

Keywords

Links

Crossrefs

Cf. A001156, A218481, A266232, A294500, A294530.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[1/(1 - x^(k^2)), {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) * A001156(k).
a(n) ~ exp(3 * 2^(-5/3) * Pi^(1/3) * Zeta(3/2)^(2/3) * n^(1/3)) * Zeta(3/2)^(2/3) * 2^(n - 7/6) / (sqrt(3) * Pi^(7/6) * n^(7/6)).
G.f.: (1/(1 - x))*Product_{k>=1} 1/(1 - x^(k^2)/(1 - x)^(k^2)). - Ilya Gutkovskiy, Aug 20 2018

A294530 Binomial transform of A023871.

Original entry on oeis.org

1, 2, 8, 33, 131, 497, 1834, 6635, 23622, 82942, 287656, 986552, 3349165, 11263951, 37558235, 124240204, 407951848, 1330340478, 4310385956, 13881618570, 44451643311, 141578435571, 448634389388, 1414774796929, 4441038400458, 13879652908322, 43197263002063
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 02 2017

Keywords

Links

Crossrefs

Cf. A023871, A218481, A266232, A294500, A294529.

Programs

  • Mathematica
    nmax = 40; s = CoefficientList[Series[Product[1/(1 - x^k)^(k^2), {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) * A023871(k).
a(n) ~ exp(2^(5/4) * 3^(-5/4) * 5^(-1/4) * Pi * n^(3/4) + Pi^2 * sqrt(n) / (4*sqrt(30)) - Pi^3 * n^(1/4) / (32 * 2^(1/4) * 15^(3/4)) + Pi^4/3840 - Zeta(3)/(4*Pi^2)) * 2^(n - 7/8) / (15^(1/8) * n^(5/8)).
G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_3(k)*x^k/(k*(1 - x)^k)). - Ilya Gutkovskiy, Aug 20 2018
Showing 1-9 of 9 results.

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