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 1-2 of 2 results.

A277374 a(n) = 2^n*floor(n/2)!*Gamma(ceiling((n+1)/2),-1/4)*exp(-1/4).

Original entry on oeis.org

1, 2, 3, 6, 50, 100, 1794, 3588, 114840, 229680, 11483880, 22967760, 1653679440, 3307358880, 324121165200, 648242330400, 82975018331520, 165950036663040, 26883905939049600, 53767811878099200, 10753562375623468800, 21507124751246937600, 5204724189801718982400
Offset: 0

Views

Author

Peter Luschny, Oct 17 2016

Keywords

Links

Crossrefs

a(n) * A056040(n) = A277393(n).

Programs

  • Maple
    a := n -> 2^n*floor(n/2)!*GAMMA(ceil((n+1)/2),-1/4)*exp(-1/4):
seq(simplify(a(n)),n=0..22);
  • Mathematica
    a[n_] := 2^n Floor[n/2]! Gamma[Ceiling[(1 + n)/2],-1/4] Exp[-1/4];
FunctionExpand@Table[a[n], {n,0,22}]
  • PARI
    for(n=0, 20, print1(round(2^n*(floor(n/2))!*exp(-1/4)* incgam(ceil((n +1)/2), -1/4)), ", ")) \\ G. C. Greubel, May 16 2018

Formula

a(n) ~ Pi*exp(-1/4-n)*n^(n+1) if n is even, a(n) ~ 2*Pi*exp(3/4-n)*(n-1)^n if n is odd. - Vladimir Reshetnikov, Oct 19 2016

A277472 a(n) = (-i)^n * Integral_{x>=0} H_n(i*x) * exp(-x), where H_n(x) is n-th Hermite polynomial, i=sqrt(-1).

Original entry on oeis.org

1, 2, 10, 60, 492, 4920, 59160, 828240, 13253520, 238563360, 4771297440, 104968543680, 2519245713600, 65500388553600, 1834010896798080, 55020326903942400, 1760650461445075200, 59862115689132556800, 2155036164826415270400, 81891374263403780275200
Offset: 0

Views

Author

Vladimir Reshetnikov, Oct 16 2016

Keywords

Links

Crossrefs

Cf. A056545, A277393, A277374.

Programs

  • Mathematica
    FunctionExpand@Table[Exp[1/4] (-2 I)^n n! (Cos[Pi n/2] Gamma[n/2 + 1, 1/4]/Gamma[n/2 + 1] + I Gamma[(n + 1)/2, 1/4] Sin[Pi n/2]/Gamma[(n + 1)/2]), {n, 0, 20}]
FunctionExpand@Table[2^n (n!/Floor[n/2]!) Gamma[Ceiling[(n+1)/2], 1/4] Exp[1/4], {n, 0, 19}] (* Peter Luschny, Oct 19 2016 *)
  • PARI
    for(n=0, 30, print1(round(2^n*(n!/floor(n/2)!)* incgam(ceil( (n+1)/2), 1/4)*exp(1/4)), ", ")) \\ G. C. Greubel, Jul 12 2018
  • Sage
    def A():
    yield 1
    yield 2
    a, h, f, g, n, b = 10, 5, 1, 2, 2, False
    while True:
        yield a
        if b:
            f = h
            h = 4 * n * h + 1
            n += 1
            a = (a * h) // f
        else:
            g += 4
            a *= g
        b = not b
a = A(); print([next(a) for  in range(20)]) # _Peter Luschny, Oct 19 2016

Formula

a(n) = exp(1/4)*(-2*i)^n * n!*( cos(Pi*n/2)*Gamma(n/2 +1, 1/4)/Gamma(n/2 +1) + i*Gamma((n+1)/2, 1/4)*sin(Pi*n/2)/Gamma((n+1)/2) ).
From Peter Luschny, Oct 19 2016: (Start)
a(n) = 2^n*(n!/floor(n/2)!)*Gamma(ceiling((n+1)/2),1/4)*exp(1/4).
The swinging factorial A056040(n) divides a(n). (End)
Showing 1-2 of 2 results.

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