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

0,3

Links

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

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

Example

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 ]*...

Mathematica

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

Prog

(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)}

Crossrefs

Cf. A000009, A218482, A266232, A307501.

Sequence in context: A019486 A019485 A018914 * A034943 A181984 A227807

Adjacent sequences: A129516 A129517 A129518 * A129520 A129521 A129522

Keyword

nonn

Author

Paul D. Hanna, Apr 18 2007

Status

approved