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 A294500 Binomial transform of the number of planar partitions (A000219). 10
 1, 2, 6, 19, 60, 185, 559, 1662, 4875, 14134, 40564, 115370, 325465, 911355, 2534595, 7004827, 19246626, 52596377, 143006632, 386984573, 1042537831, 2796803110, 7473161196, 19893461042, 52767059608, 139488323734, 367540167625, 965445514862, 2528516552660 (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..2930 FORMULA a(n) = Sum_{k=0..n} binomial(n,k) * A000219(k). a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) - Zeta(3)/12) * 2^(n + 7/18) * Zeta(3)^(7/36) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)). - Ilya Gutkovskiy, Aug 20 2018 MATHEMATICA nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}] CROSSREFS Cf. A218481, A266232, A294501, A294502, A294504. Sequence in context: A014346 A183188 A118364 * A208481 A052544 A204200 Adjacent sequences: A294497 A294498 A294499 * A294501 A294502 A294503 KEYWORD nonn AUTHOR Vaclav Kotesovec, Nov 01 2017 STATUS approved

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Last modified September 18 02:51 EDT 2024. Contains 375995 sequences. (Running on oeis4.)