A218481 Binomial transform of the partition numbers (A000041).

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

0,2

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)

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

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

Example

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; ...

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 *)

Prog

(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)}

Crossrefs

Cf. A218482, A222115, A000041, A000203, A266497.

Cf. A294466, A281425, A095051.

Cf. A266232, A294467, A293467, A294468, A294500.

Sequence in context: A318234 A371426 A027931 * A267905 A209230 A103142

Adjacent sequences: A218478 A218479 A218480 * A218482 A218483 A218484

Keyword

nonn,nice

Author

Paul D. Hanna, Oct 29 2012

Status

approved