A218481 -id:A218481 - cp's OEIS frontend

A294502 Binomial transform of A026007.

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

Vaclav Kotesovec, Nov 01 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2960

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

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}]

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

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

Ilya Gutkovskiy, Aug 18 2018

nonn

Binomial transform of A006906.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
N. J. A. Sloane, Transforms

Cf. A006906, A218481, A294500.

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

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}]

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

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

Vaclav Kotesovec, Nov 01 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2840

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

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}]

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

Vaclav Kotesovec, Nov 02 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

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

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}]

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

A300511 Expansion of e.g.f. exp(Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).

Original entry on oeis.org

1, 1, 3, 10, 42, 203, 1119, 6841, 45916, 334414, 2622256, 21984668, 195991611, 1849158088, 18390563792, 192128761836, 2102097270199, 24022460183508, 286060559298908, 3542047217686560, 45517563689858955, 606014811356799054, 8346153294214800894, 118731713512110007282

Offset: 0

Ilya Gutkovskiy, Mar 07 2018

nonn

Exponential transform of A000041.

			E.g.f.: A(x) = 1 + x/1! + 3*x^2/2! + 10*x^3/3! + 42*x^4/4! + 203*x^5/5! + 1119*x^6/6! + 6841*x^7/7! + 45916*x^8/8! + ..

Alois P. Heinz, Table of n, a(n) for n = 0..533
N. J. A. Sloane, Transforms
Index entries for related partition-counting sequences

Cf. A000041, A001970, A058892, A215915, A218481, A300512.

a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*
      binomial(n-1, j-1)*combinat[numbpart](j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Mar 07 2018

nmax = 23; CoefficientList[Series[Exp[Sum[PartitionsP[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[PartitionsP[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

E.g.f.: exp(Sum_{k>=1} A000041(k)*x^k/k!).

A356267 a(n) = Sum_{k=0..n} binomial(2n, k) p(k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 3, 17, 97, 583, 3275, 18988, 104821, 584441, 3180889, 17295626, 92225785, 492811733, 2590911097, 13591889993, 70605682273, 365601169939, 1876312271003, 9605682510676, 48809295651049, 247315330613099, 1245888505795725, 6256686417801919, 31260996876796579

Offset: 0

Vaclav Kotesovec, Aug 01 2022

nonn

Cf. A000041, A032443, A218481, A286955, A356268.

Table[Sum[Binomial[2*n, k] * PartitionsP[k], {k, 0, n}], {n, 0, 30}]

a(n) ~ erfc(Pi/(2*sqrt(6))) * 2^(2*n - 3) * exp(Pi*sqrt(2*n/3) + Pi^2/24) / (sqrt(3)*n).

A356280 a(n) = Sum_{k=0..n} binomial(2n, n-k) p(k), where p(k) is the partition function A000041.

Original entry on oeis.org

1, 3, 12, 50, 211, 894, 3791, 16068, 68032, 287675, 1214761, 5122428, 21571028, 90718913, 381050570, 1598645263, 6699355413, 28044720813, 117281866330, 489999068614, 2045341248508, 8530263939665, 35547083083270, 148015639243691, 615870619714675, 2560734764460360

Offset: 0

Vaclav Kotesovec, Aug 01 2022

nonn

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, VI.26. Catalan sums, p.417.

Cf. A000041, A032443, A218481, A286955, A356267, A356281.

Table[Sum[PartitionsP[k]*Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]
nmax = 30; CoefficientList[Series[Sum[PartitionsP[k]*((1-2*x-Sqrt[1-4*x])/(2*x))^k / Sqrt[1-4*x], {k, 0, nmax}], {x, 0, nmax}], x]

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

A222115 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * sigma(k).

Original entry on oeis.org

2, 6, 17, 46, 117, 285, 674, 1558, 3536, 7911, 17503, 38377, 83501, 180480, 387882, 829606, 1766999, 3749766, 7931115, 16724871, 35173778, 73794661, 154485528, 322771345, 673155142, 1401536935, 2913490376, 6047714600, 12536770559, 25956242580, 53678385267, 110889844998

Offset: 1

Paul D. Hanna, Jun 01 2013

nonn

Here sigma(n) is the sum of divisors of n (A000203).

			L.g.f.: L(x) = 2*x + 6*x^2/2 + 17*x^3/3 + 46*x^4/4 + 117*x^5/5 + 285*x^6/6 +...
where
exp(L(x)) = 1 + 2*x + 5*x^2 + 13*x^3 + 34*x^4 + 88*x^5 + 225*x^6 + 569*x^7 +...+ A218481(n)*x^n +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A218481, A185003, A000203.

Table[Sum[Binomial[n,k]DivisorSigma[1,k],{k,n}],{n,40}]+1 (* Harvey P. Dale, Jul 21 2015 *)

{a(n)=1+sum(k=1,n,binomial(n,k)*sigma(k))}
for(n=1,30,print1(a(n),", "))

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(m=1, n+1, x^m/((1-x)^m-X^m)/m), n)}

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(k=1, n, k*log(1-X)-log((1-x)^k-X^k)), n)}

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m), n)}

{a(n)=local(X=x+x*O(x^n)); n*polcoeff(-log(1-X)+sum(k=1, n, valuation(2*k, 2)*log(1 + x^k/(1-X)^k)), n)}

Logarithmic derivative of the binomial transform of the partition numbers (A218481).
L.g.f.: -log(1-x) + Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.
L.g.f.: -log(1-x) + Sum_{n>=1} x^n/((1-x)^n - x^n) / n.
L.g.f.: -log(1-x) + Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).
L.g.f.: -log(1-x) + Sum_{n>=1} A001511(n) * log(1 + x^n/(1-x)^n), where 2^A001511(n) is the highest power of 2 that divides 2*n.
a(n) = A185003(n) + 1.
a(n) ~ Pi^2/12 * n * 2^n. - Vaclav Kotesovec, Dec 30 2015

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

Vaclav Kotesovec, Nov 02 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..2600

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

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}]

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

A307755 Exponential convolution of partition numbers (A000041) with themselves.

Original entry on oeis.org

1, 2, 6, 18, 58, 184, 586, 1822, 5618, 16980, 50892, 150064, 439210, 1268924, 3640342, 10337596, 29160638, 81570368, 226795202, 626070664, 1718783084, 4689582366, 12730998988, 34373603158, 92385339242, 247099560046, 658137847408, 1745322097886, 4610549234836, 12131656526628

Offset: 0

Ilya Gutkovskiy, Apr 26 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A000041, A000712, A218481, A307756.

a:= n-> (p-> add(binomial(n, j)*p(j)*p(n-j), j=0..n))(combinat[numbpart]):
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 26 2019

nmax = 29; CoefficientList[Series[Sum[PartitionsP[k] x^k/k!, {k, 0, nmax}]^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] PartitionsP[k] PartitionsP[n - k], {k, 0, n}], {n, 0, 29}]

E.g.f.: (Sum_{k>=0} A000041(k)*x^k/k!)^2.
a(n) = Sum_{k=0..n} binomial(n,k)*A000041(k)*A000041(n-k).
a(n) ~ exp(2*Pi*sqrt(n/3)) * 2^(n-2) / (3*n^2). - Vaclav Kotesovec, May 06 2019

cp's OEIS Frontend

A294502 Binomial transform of A026007.

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

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A294504 Binomial transform of A156616.

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A294529 Binomial transform of A001156.

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A300511 Expansion of e.g.f. exp(Sum_{k>=1} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).

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A356267 a(n) = Sum_{k=0..n} binomial(2*n, k) * p(k), where p(k) is the partition function A000041.

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A356280 a(n) = Sum_{k=0..n} binomial(2*n, n-k) * p(k), where p(k) is the partition function A000041.

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A222115 a(n) = 1 + Sum_{k=1..n} binomial(n,k) * sigma(k).

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A356267 a(n) = Sum_{k=0..n} binomial(2n, k) p(k), where p(k) is the partition function A000041.

A356280 a(n) = Sum_{k=0..n} binomial(2n, n-k) p(k), where p(k) is the partition function A000041.