A348208 a(n) = Sum_{k=0..floor(n/2)} (-1)^(k-1)*(k-1)^2*A106828(n, k).
-1, 0, 0, 0, -3, -20, -70, -84, 1267, 18824, 209484, 2284920, 26010369, 314864628, 4073158102, 56304102596, 830061867975, 13016975343184, 216535182535928, 3810394068301296, 70744547160678501, 1382375535029293500, 28364229790262962386, 609820072529413714012
Offset: 0
Keywords
Examples
E.g.f.: -1 - 3*x^4/4! - 20*x^5/5! - 70*x^6/6! - 84*x^7/7! + 1267*x^8/8! + 18824*x^9/9! + ...
a(11) = Sum_{k=0..5} (-1)^(k-1)*(k-1)^2*A106828(11, k).
a(11) = (-1)*1*0 + (1)*0*3628800 + (-1)*1*6636960 + (1)*4*3678840 + (-1)*9*705320 + (1)*16*34650 = 2284920.
For k = 0, A106828(11,0) = 0.
For k = 1, (1-1)^2 = 0.
For 2 <= k <= 5, A106828(11, k) == 0 (mod 11*10).
Result a(11) == 0 (mod 11*10).
Programs
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Maple
a := series((-1+2*x-2*x^2+x^3+(1-x)*(log((1-x)^(1-2*x))-(log(1-x))^2))*exp(x), x=0, 24): seq(n!*coeff(a, x, n), n=0..23); # second program: a := n -> add((-1)^(k-1)*(k-1)^2*A106828(n, k), k=0..iquo(n, 2)): seq(a(n), n=0..23);
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Mathematica
CoefficientList[Series[(-1+2*x-2*x^2+x^3+(1-x)*(Log[(1-x)^(1-2*x)]-(Log[1-x])^2))*Exp[x], {x, 0, 23}], x]*Range[0, 23]! -
PARI
my(x='x+O('x^30)); Vec(serlaplace((-1 + 2*x - 2*x^2 + x^3 + (1 - x)*(log((1 - x)^(1 - 2*x)) - (log(1 - x))^2))*exp(x))) \\ Michel Marcus, Oct 07 2021
Formula
E.g.f.: (-1 + 2*x - 2*x^2 + x^3 + (1 - x)*(log((1 - x)^(1 - 2*x)) - (log(1 - x))^2))*exp(x).
a(n) ~ 2 * exp(1) * log(n) * n! / n^2 * (1 + (gamma - 3/2)/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 09 2021
Comments