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.

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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

A185003 a(n) = Sum_{k=1..n} binomial(n,k)*sigma(k).

Original entry on oeis.org

1, 5, 16, 45, 116, 284, 673, 1557, 3535, 7910, 17502, 38376, 83500, 180479, 387881, 829605, 1766998, 3749765, 7931114, 16724870, 35173777, 73794660, 154485527, 322771344, 673155141, 1401536934, 2913490375, 6047714599, 12536770558, 25956242579, 53678385266
Offset: 1

Views

Author

Paul D. Hanna, Feb 04 2012

Keywords

Examples

			L.g.f.: L(x) = x + 5*x^2/2 + 16*x^3/3 + 45*x^4/4 + 116*x^5/5 + 284*x^6/6 +...
where exponentiation yields A103446 (with offset=0):
exp(L(x)) = 1 + x + 3*x^2 + 8*x^3 + 21*x^4 + 54*x^5 + 137*x^6 + 344*x^7 +...

Links

Crossrefs

Cf. A103446, A222115, A000203, A160399.

Programs

  • Magma
    [&+[Binomial(n,k)*DivisorSigma(1,k):k in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019
  • Magma
    [&+[&+[i*Binomial(n,i*j):j in [1..n]]:i in [1..n]]:n in [1..31]]; // Marius A. Burtea, Nov 12 2019
  • Maple
    with(numtheory): seq(add(binomial(n,i)*sigma(i), i=1..n), n=1..40); # Ridouane Oudra, Nov 12 2019
  • Mathematica
    Table[Sum[Binomial[n, k] DivisorSigma[1, k], {k, n}], {n,50}] (* G. C. Greubel, Jun 03 2017 *)
  • PARI
    {a(n)=sum(k=1,n,sigma(k)*binomial(n,k))}
for(n=1,30,print1(a(n),", "))
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(m=1, n+1, x^m/((1-x)^m-X^m)/m), n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(k=1, n, k*log(1-X)-log((1-x)^k-X^k)), n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(m=1, n+1, sigma(m)*x^m/(1-X)^m/m), n)}
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n*polcoeff(sum(k=1, n, valuation(2*k, 2)*log(1 + x^k/(1-X)^k)), n)}

Formula

Logarithmic derivative of A103446 (with offset=0), which describes the binomial transform of partitions.
From Paul D. Hanna, Jun 01 2013: (Start)
L.g.f.: Sum_{n>=1} sigma(n) * x^n/(1-x)^n / n.
L.g.f.: Sum_{n>=1} x^n/((1-x)^n - x^n) / n.
L.g.f.: Sum_{n>=1} n*log(1-x) - log((1-x)^n - x^n).
L.g.f.: 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) = A222115(n) - 1. (End)
a(n) ~ Pi^2/12 * n * 2^n. - Vaclav Kotesovec, Dec 30 2015
a(n) = Sum_{i=1..n} Sum_{j=1..n} i*binomial(n,i*j). - Ridouane Oudra, Nov 12 2019
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