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 1-6 of 6 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

A294499 Inverse binomial transform of the number of overpartitions (A015128).

Original entry on oeis.org

1, 1, 1, 1, -1, 3, -5, 7, -7, 3, 5, -9, -17, 129, -417, 977, -1809, 2591, -2317, -1061, 10485, -27983, 49165, -51319, -26861, 311455, -1011473, 2393275, -4643591, 7521265, -9694135, 7738137, 4976985, -38789975, 106112817, -215068927, 354515933, -464539803
Offset: 0

Views

Author

Vaclav Kotesovec, Nov 01 2017

Keywords

Links

Crossrefs

Cf. A266497, A281425, A293467.

Programs

  • Mathematica
    Table[Sum[(-1)^(n-k)*Binomial[n, k]*Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}], {k, 0, n}], {n, 0, 50}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A015128(k).
a(n) = [x^n] (1 - x)^n/theta_4(x), where theta_4() is the Jacobi theta function. - Ilya Gutkovskiy, Nov 03 2017

A318570 Expansion of Product_{k>=1} ((1 - x)^k + x^k)/((1 - x)^k - x^k).

Original entry on oeis.org

1, 2, 6, 18, 52, 146, 402, 1090, 2916, 7708, 20160, 52236, 134222, 342304, 867024, 2182384, 5461696, 13595918, 33677550, 83036878, 203859820, 498470998, 1214230586, 2947204870, 7129403128, 17191258642, 41328057106, 99067295658, 236822823336, 564650823162, 1342921372126
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 15 2018

Keywords

Comments

First differences of the binomial transform of A015128.
Convolution of A129519 and A218482.

Links

Crossrefs

Cf. A000203, A015128, A129519, A218482, A266497.

Programs

  • Maple
    a:=series(mul(((1-x)^k+x^k)/((1-x)^k-x^k),k=1..100),x=0,31): seq(coeff(a,x,n),n=0..30); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 30; CoefficientList[Series[Product[((1 - x)^k + x^k)/((1 - x)^k - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[1/EllipticTheta[4, 0, x/(1 - x)], {x, 0, nmax}], x]
nmax = 30; CoefficientList[Series[Exp[Sum[(DivisorSigma[1, 2 k] - DivisorSigma[1, k]) x^k/(k (1 - x)^k), {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

G.f.: 1/theta_4(x/(1 - x)), where theta_4() is the Jacobi theta function.
G.f.: exp(Sum_{k>=1} (sigma(2*k) - sigma(k))*x^k/(k*(1 - x)^k)).
a(n) ~ 2^(n-3) * exp(Pi*sqrt(n/2) + Pi^2/16) / n. - Vaclav Kotesovec, Oct 15 2018

A307945 Exponential convolution of A015128 with themselves.

Original entry on oeis.org

1, 4, 16, 64, 252, 968, 3616, 13120, 46432, 160772, 545856, 1821056, 5979520, 19350552, 61795968, 194964672, 608261628, 1878140024, 5743681784, 17408223328, 52320105080, 156011658272, 461763417056, 1357182242560, 3962591708576, 11497241014652
Offset: 0

Views

Author

Vaclav Kotesovec, May 07 2019

Keywords

Links

Crossrefs

Cf. A015128, A266497, A294499, A307755, A307756.

Programs

  • Maple
    S:= series(1/JacobiTheta4(0,q),q,101):
f:= n -> add(binomial(n,k)*coeff(S,q,k)*coeff(S,q,n-k),k=0..n):
map(f, [$0..100]); # Robert Israel, May 08 2019
  • Mathematica
    A015128[n_]:=Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}]; Table[Sum[Binomial[n, k]*A015128[k]*A015128[n-k], {k, 0, n}], {n, 0, 25}]

Formula

a(n) = Sum_{k=0..n} binomial(n,k) * A015128(k) * A015128(n-k).
a(n) ~ 2^(n-4) * exp(Pi*sqrt(2*n)) / n^2.

A356289 a(n) = Sum_{k=0..n} binomial(2*n, n-k) * v(k), where v(k) is the number of overpartitions of n (A015128).

Original entry on oeis.org

1, 4, 18, 82, 372, 1676, 7500, 33358, 147570, 649722, 2848524, 12441434, 54155774, 235008672, 1016971480, 4389589484, 18902538548, 81222609020, 348308661820, 1490884718484, 6370468593732, 27176620756392, 115760526170340, 492386739902574, 2091554077819948, 8873225318953248
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2022

Keywords

Crossrefs

Cf. A015128, A266497, A356280, A356281, A356282, A356283, A356290.

Programs

  • Mathematica
    Table[Sum[Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}] * Binomial[2*n, n-k], {k, 0, n}], {n, 0, 30}]

Formula

a(n) ~ 2^(2*n - 7/6) * exp(3 * Pi^(4/3) * n^(1/3) / 2^(8/3)) / (sqrt(3) * Pi^(2/3) * n^(2/3)).

A356290 a(n) = Sum_{k=0..n} binomial(3*n, n-k) * v(k), where v(k) is the number of overpartitions of n (A015128).

Original entry on oeis.org

1, 5, 31, 200, 1309, 8627, 57082, 378648, 2516111, 16740913, 111494801, 743137984, 4956359312, 33074272702, 220810039566, 1474764797488, 9853307017341, 65853733243281, 440255398634199, 2944041287677060, 19691951641479427, 131744163990056479, 881586559906575688
Offset: 0

Views

Author

Vaclav Kotesovec, Aug 02 2022

Keywords

Crossrefs

Cf. A015128, A266497, A356280, A356281, A356282, A356283, A356289.

Programs

  • Mathematica
    Table[Sum[Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}] * Binomial[3*n, n-k], {k, 0, n}], {n, 0, 30}]

Formula

a(n) ~ c * 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n + 1)), where c = Sum_{j>=0} v(j)/2^j = 8.2559879357782500655441408494322731265270016167882303456037...
Showing 1-6 of 6 results.

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