cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A295098 a(n) = n! * [x^n] exp(n*x)*(1 + exp(x^2/2)*x*(1 + sqrt(Pi/2)*erf(x/sqrt(2)))).

Original entry on oeis.org

1, 2, 10, 75, 760, 9715, 150060, 2719017, 56556480, 1328337117, 34773226340, 1003998156293, 31696623421488, 1086258754644505, 40161805428662876, 1593475984997421525, 67534151717002711296, 3044989873158805787409, 145537456143562934305860, 7350253384336351186239341, 391132792671917087054081200
Offset: 0

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Author

Ilya Gutkovskiy, Nov 14 2017

Keywords

Comments

The n-th term of the n-th binomial transform of A006882.

Links

Crossrefs

Cf. A006882, A063170, A263529, A295099, A295100.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x] (1 + Exp[x^2/2] x (1 + Sqrt[Pi/2] Erf[x/Sqrt[2]])), {x, 0, n}], {n, 0, 20}]

Formula

a(n) ~ c * n^n, where c = 1 + exp(1/2) * (1 + sqrt(Pi/2) * erf(1/sqrt(2))) = 4.0594074053425761445394754992332... - Vaclav Kotesovec, Aug 21 2018

A262020 Inverse binomial transform of double factorial n!! = A006882(n).

Original entry on oeis.org

1, 0, 1, -1, 5, -11, 43, -127, 489, -1693, 6771, -26071, 109693, -457757, 2028671, -9039931, 42101329, -198411489, 967906675, -4791497559, 24401815141, -126243354637, 669094876055, -3603105436163, 19818039219577, -110721426757801, 630419303537115
Offset: 0

Views

Author

Alois P. Heinz, Oct 22 2015

Keywords

Links

Crossrefs

Cf. A000166 (the same for n!), A006882, A263529.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n<3, (n-1)^2,
       (n-2)*a(n-3) +(n-1)*a(n-2) -2*a(n-1))
    end:
seq(a(n), n=0..30);
  • Mathematica
    Table[Sum[(-1)^(n-k) * Binomial[n, k] * k!!, {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Oct 31 2017 *)

Formula

E.g.f.: (2*exp(x^2/2)*x+2+sqrt(2*Pi)*exp(x^2/2)*erf(x/sqrt(2))*x) / (2*exp(x)).
a(n) = (n-2)*a(n-3)+(n-1)*a(n-2)-2*a(n-1) for n>2, a(n) = (n-1)^2 otherwise.
a(n) = Sum_{k=0..n} (-1)^k * C(n,k) * A006882(n-k).
a(n) ~ (-1)^n * (sqrt(Pi) - sqrt(2)) * exp(sqrt(n) - n/2 - 1/4) * n^((n+1)/2) / 2. - Vaclav Kotesovec, Oct 31 2017
G.f.: Sum_{k>=0} k!!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019
Showing 1-2 of 2 results.

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