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 A129519 First differences of the binomial transform of the distinct partition numbers (A000009). 13
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 23 23:00 EDT 2021. Contains 345402 sequences. (Running on oeis4.)