A129519 First differences of the binomial transform of the distinct partition numbers (A000009).

1, 1, 2, 5, 12, 28, 65, 151, 350, 807, 1850, 4221, 9597, 21760, 49215, 111032, 249856, 560835, 1255854, 2805969, 6256784, 13925698, 30941050, 68634679, 152009239, 336152787, 742276931, 1636747349, 3604206106, 7926412320, 17410413153

Offset: 0

Author

Paul D. Hanna, Apr 18 2007

Keywords

• nonn

Examples

			Product formula is illustrated by:
A(x) = [1 + x + x^2 + x^3 + x^4 + x^5 +...]*
[1 + x^2 + 2x^3 + 3x^4 + 4x^5 + 5x^6 +...]*
[1 + x^3 + 3x^4 + 6x^5 + 10x^6 + 15x^7 +...]*
[1 + x^4 + 4x^5 + 10x^6 + 20x^7 + 35x^8 +...]*
[1 + x^5 + 5x^6 + 15x^7 + 35x^8 + 70x^9 +...]*...*
[1 + Sum_{k>=n+1} C(k-1,n)*x^k ]*...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000009, A218482, A266232, A307501.

Programs

Mathematica

Flatten[{1, Differences[Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 40}]]}] (* Vaclav Kotesovec, Oct 30 2017 *)

PARI

{a(n)=polcoeff(prod(k=0,n,1+sum(i=k+1,n,binomial(i-1,k)*x^i +x*O(x^n))),n)}

Formula

G.f.: A(x) = Product_{n>=1} [1 + x^n/(1-x)^n].

a(n) = A266232(n) - A266232(n-1), for n>0. - Vaclav Kotesovec, Oct 30 2017

a(n) ~ exp(Pi*sqrt(n/6) + Pi^2/48) * 2^(n - 9/4) / (3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 30 2017