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 A266232 Binomial transform of the number of partitions into distinct parts (A000009). 21
 1, 2, 4, 9, 21, 49, 114, 265, 615, 1422, 3272, 7493, 17090, 38850, 88065, 199097, 448953, 1009788, 2265642, 5071611, 11328395, 25254093, 56195143, 124829822, 276839061, 612991848, 1355268779, 2992016128, 6596222234, 14522634554, 31933047707, 70130243427 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..3200 FORMULA a(n) ~ 2^(n-5/4) * exp(Pi*sqrt(n/6) + Pi^2/48) / (3^(1/4)*n^(3/4)). G.f.: (1/(1 - x))*Product_{k>=1} (1 + x^k/(1 - x)^k). - Ilya Gutkovskiy, Aug 19 2018 MATHEMATICA Table[Sum[Binomial[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 50}] nmax = 30; CoefficientList[Series[Sum[PartitionsQ[k] * x^k / (1-x)^(k+1), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 31 2022 *) CROSSREFS Cf. A000009, A294467, A293467, A294468. Cf. A218481, A294466, A281425, A095051, A294500. Sequence in context: A281425 A101891 A119967 * A052921 A219150 A322325 Adjacent sequences: A266229 A266230 A266231 * A266233 A266234 A266235 KEYWORD nonn AUTHOR Vaclav Kotesovec, Dec 25 2015 STATUS approved

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Last modified September 19 13:45 EDT 2024. Contains 376012 sequences. (Running on oeis4.)