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.

Showing 1-3 of 3 results.

A084261 A binomial transform of factorial numbers.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 52, 134, 361, 1009, 2926, 8768, 27121, 86373, 282864, 950866, 3277169, 11564353, 41739130, 153919324, 579411641, 2224535125, 8703993420, 34681783422, 140637608089, 580019801201, 2431509498406, 10355296410712
Offset: 0

Views

Author

Paul Barry, May 26 2003

Keywords

Comments

Binomial transform of A000142 (with interpolated zeros).
Row sums of A161556. Hankel transform is A137704. [Paul Barry, Apr 11 2010]

Links

Programs

  • Mathematica
    Table[Sum[Binomial[n,2*k]*k!, {k,0,Floor[n/2]}], {n,0,50}] (* G. C. Greubel, Jan 24 2017 *)
  • PARI
    for(n=0,50, print1(sum(k=0,floor(n/2), binomial(n,2*k)*k!), ", ")) \\ G. C. Greubel, Jan 24 2017

Formula

a(n) = Sum_{k=0..floor(n/2)} C(n, 2k)*k!.
a(n) = Sum_{k=0..n} C(n, k)*(k/2)!*((1+(-1)^k)/2) .
E.g.f.: exp(x)*(1+sqrt(Pi)/2*x*exp(x^2/4)*erf(x/2)). - Vladeta Jovovic, Sep 25 2003
O.g.f.: A(x) = 1/(1-x-x^2/(1-x-x^2/(1-x-2*x^2/(1-x-2*x^2/(1-x-3*x^2/(1-... -x-[(n+1)/2]*x^2/(1- ...))))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
a_n ~ (1/2) * sqrt(Pi*n/e)*(n/2)^(n/2)*exp(-n/2 + sqrt(2n)). - Cecil C Rousseau (ccrousse(AT)memphis.edu), Mar 14 2006: (cf. A002896).
Conjecture: 2*a(n) -4*a(n-1) +(-n+2)*a(n-2) +(n-2)*a(n-3)=0. - R. J. Mathar, Nov 30 2012

A162748 Row sums of factorial-Pascal matrix A162747.

Original entry on oeis.org

1, 2, 5, 14, 42, 132, 430, 1444, 4984, 17648, 64024, 237712, 902416, 3499680, 13853424, 55931168, 230142848, 964460288, 4113656704, 17846729984, 78708574976, 352678567424, 1604739694848, 7411167960576, 34723660917760
Offset: 0

Views

Author

Paul Barry, Jul 12 2009

Keywords

Comments

Second binomial transform of aerated factorial numbers. Binomial transform of A084261. Hankel transform is A137704.

Links

Programs

  • Mathematica
    Table[Sum[Binomial[n,k]*2^(n-k)*(k/2)!*(1+(-1)^k)/2,{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Aug 15 2013 *)

Formula

G.f.: 1/(1-2x-x^2/(1-2x-x^2/(1-2x-2x^2/(1-2x-2x^2/(1-2x-3x^2/(1-2x-3x^2/(1-2x-4x^2/(1-2x-... (continued fraction);
a(n)=sum{k=0..floor(n/2), C(n,2k)*2^(n-2k)*F(k+1)}=sum{k=0..n, C(n,k)*2^(n-k)*(k/2)!*(1+(-1)^k)/2}.
a(n)=sum{k=0..n, A161556(n,k)*2^k}. - Paul Barry, Apr 11 2010
E.g.f.: exp(2x)*(1+(sqrt(Pi)/2)*x*exp(x^2/4)*erf(x/2)). - Paul Barry, Sep 17 2010
Apparently -2*a(n) +8*a(n-1) +(n-8)*a(n-2) +2*(2-n)*a(n-3)=0. - R. J. Mathar, Oct 25 2012
a(n) ~ 1/2 * sqrt(Pi*n) * exp(2*sqrt(2*n)-n/2-2) * (n/2)^(n/2) * (1 + 1/(3*sqrt(2*n))). - Vaclav Kotesovec, Aug 15 2013

Extensions

Minor edits by Vaclav Kotesovec, Jul 22 2015

A361522 The aerated factorial numbers.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 6, 0, 24, 0, 120, 0, 720, 0, 5040, 0, 40320, 0, 362880, 0, 3628800, 0, 39916800, 0, 479001600, 0, 6227020800, 0, 87178291200, 0, 1307674368000, 0, 20922789888000, 0, 355687428096000, 0, 6402373705728000, 0, 121645100408832000, 0, 2432902008176640000
Offset: 0

Views

Author

Peter Luschny, Mar 14 2023

Keywords

Comments

An aerated version of A000142, which is the main entry for this sequence.

Links

Crossrefs

Cf. A000142, A081124, A084261, A081123, A161556.

Programs

  • Maple
    egf := (z/2)*Pi^(1/2)*erf(z/2)*exp((z/2)^2) + 1:
ser := series(egf, z, 42): seq(n!*coeff(ser, z, n), n = 0..40);
  • Mathematica
    a[n_] := If[OddQ[n], 0, (n/2)!]; Array[a, 41, 0] (* Amiram Eldar, Mar 14 2023 *)

Formula

a(n) = n! * [z^n] (z/2)*Pi^(1/2)*erf(z/2)*exp((z/2)^2) + 1.
a(n) = n! * [z^n] 1 + 2*u*exp(u)*hypergeom([1/2], [3/2], -u), where u = (z/2)^2.
Showing 1-3 of 3 results.

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