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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A203517 a(n) = A203516(n)/A000178(n).

Original entry on oeis.org

1, 4, 96, 15360, 17203200, 138726604800, 8203736501452800, 3603868630142209228800, 11873738053102139590311936000, 295578185800614925763054760099840000, 55920479534877093093661639943174183976960000
Offset: 1

Views

Author

Clark Kimberling, Jan 03 2012

Keywords

Links

Crossrefs

Cf. A000178, A000217, A034910, A093883, A203516.

Programs

  • Magma
    [2^Binomial(n,2)*(&*[Binomial(2*k,k): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Feb 19 2024
  • Mathematica
    f[j_] := 2 j - 1; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]   (* A000178 *)
Table[v[n], {n, 1, z}]                  (* A203516 *)
Table[v[n + 1]/(4 v[n]), {n, 1, z - 1}] (* A034910 *)
Table[v[n]/d[n], {n, 1, 20}]            (* A203517 *)
Table[2^(-1/24 - 3*n/2 + 3*n^2/2) * Glaisher^(3/2) * Pi^(1/4 - n/2) * BarnesG[1/2 + n] / E^(1/8) / BarnesG[1 + n], {n, 1, 12}] (* Vaclav Kotesovec, Sep 01 2023 *)
  • SageMath
    [2^binomial(n,2)*product(binomial(2*k,k) for k in range(n)) for n in range(1,21)] # G. C. Greubel, Feb 19 2024

Formula

a(n) ~ A^(3/2) * 2^(-7/24 - 3*n/2 + 3*n^2/2) * exp(-1/8 + n/2) * n^(1/8 - n/2) / Pi^(n/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 01 2023
a(n) = 2^binomial(n,2) * Product_{j=0..n-1} binomial(2*j, j). - G. C. Greubel, Feb 19 2024

A282564 Real part of A000178(n) * Sum_{k=0..n} i^k/k!, where i = sqrt(-1).

Original entry on oeis.org

1, 1, 1, 6, 156, 18720, 13443840, 67756953600, 2732085780480000, 991419288020582400000, 3597660477435617162035200000, 143607093745702043133526671360000000, 68788027941331539080620236035063808000000000, 428344480781652673551035086691251861743206400000000000
Offset: 0

Views

Author

Daniel Suteu, Feb 18 2017

Keywords

Examples

			For n = 4, a(4) = 156, which is the real part of A000178(4)*(1/0! + i/1! - 1/2! - i/3! + 1/4!) = 156+240*i.

Links

Crossrefs

The corresponding imaginary part is A282567.
Cf. A000178, A281964.

Programs

  • PARI
    a(n) = real(prod(k=0, n, k!) * sum(k=0, n, I^k/k!));

Formula

a(n) ~ cos(1) * A000178(n).
a(0) = 1, a(n) = n!*a(n-1) + A000178(n-1)*cos(Pi/2*n).
Lim_{n->infinity} a(n)/G(n+2) = cos(1), where G(z) is the Barnes G-function.

A282567 Imaginary part of A000178(n) * Sum_{k=0..n} i^k/k!, where i = sqrt(-1).

Original entry on oeis.org

0, 1, 2, 10, 240, 29088, 20943360, 105529651200, 4254955536384000, 1544043321627770880000, 5603024405522854969344000000, 223654797931768113135574056960000000, 107131006056993617020920990202331136000000000, 667107003169139201955908457896071963607040000000000000
Offset: 0

Views

Author

Daniel Suteu, Feb 18 2017

Keywords

Examples

			For n = 4, a(4) = 240, which is the imaginary part of A000178(4)*(1/0! + i/1! - 1/2! - i/3! + 1/4!) = 156+240*i.

Links

Crossrefs

The corresponding real part is A282564.
Cf. A000178, A282132.

Programs

  • PARI
    a(n) = imag(prod(k=0, n, k!) * sum(k=0, n, I^k/k!));

Formula

a(n) ~ sin(1) * A000178(n).
a(0) = 0, a(n) = n!*a(n-1) + A000178(n-1)*sin(Pi/2*n).
Lim_{n->infinity} a(n)/G(n+2) = sin(1), where G(z) is the Barnes G-function.

A379742 a(n) is the number of divisors of A000178(n).

Original entry on oeis.org

1, 1, 2, 6, 18, 72, 234, 1088, 3600, 10416, 28080, 124080, 387828, 1921024, 6926400, 20941344, 54934880, 251328000, 810152280, 4254092800, 15266200950, 46208448000, 129674387920, 640501862400, 2197261252368, 6404827161600, 17436935577600, 43314231340800, 101062601640000, 436914124416000, 1355859833328000, 7074064925491200
Offset: 0

Views

Author

Tsuyoshi Hanatate, Dec 31 2024

Keywords

Links

Crossrefs

Cf. A000005, A000178, A027423.

Programs

  • Mathematica
    DivisorSigma[0, FoldList[Times, Range[0, 30]!]] (* Paolo Xausa, Jan 06 2025 *)
  • PARI
    a(n)=numdiv(prod(k=1,n,k!));
vector(30,n,a(n-1)) \\ Joerg Arndt, Jan 02 2025
  • PARI
    a(n) = {my(prd = 1); forprime(p = 2, n, prd *= (1 + (n*(n+1)/2 - 1 - sum(i = 2, n, sumdigits(i, p)))/(p-1))); prd;} \\ Amiram Eldar, Jan 15 2025

Formula

a(n) = A000005(A000178(n)).
a(n) = Product_{primes p <= n} (1 + (n*(n+1)/2 - 1 - Sum_{i=2..n} s_p(i))/(p-1)), where s_p(i) is the sum of digits of i in base p. - Amiram Eldar, Jan 15 2025

A089501 Built from superfactorials A000178.

Original entry on oeis.org

1, 6, 2880, 870912000, 637129677864960000, 3076276241856388273274880000000, 218470761021769399142244567460557619200000000000, 444747235963340607791337561259087696911923105885061120000000000000000
Offset: 0

Views

Author

Wolfdieter Lang, Nov 07 2003

Keywords

Comments

a(n) appears as a numerator in A089500.

Formula

N(n) := sfac(n-1)*sfac(2*n+1)/sfac(n+1) with sfac(n) := product(k!, k=1..n), n>=1, sfac(0) := 1. sfac(n)= A000178(n).
a(n) ~ 2^(2*n^2 + 5*n + 23/12) * n^(2*n^2 + 2*n -1/12) * Pi^n / (A * exp(3*n^2 + 2*n - 1/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

A107251 Supercatalan numbers SF(2n)/(SF(n)*SF(n+1)) where SF is the superfactorial function A000178.

Original entry on oeis.org

1, 1, 12, 7200, 508032000, 7742895390720000, 40797452088662556672000000, 108985983996792124183843071590400000000, 203800994173724454677862841368011757060096000000000000
Offset: 0

Views

Author

Henry Bottomley, May 14 2005

Keywords

Examples

			a(3) = 1!*2!*3!*4!*5!*6!/(1!*2!*3!*1!*2!*3!*4!) = 24883200/(12*288) = 7200.

Crossrefs

Cf. A000108 for original Catalan numbers (2n)!/(n!*(n+1)!).
Cf. A000142, A000178, A079478.

Programs

  • Maple
    seq(mul(mul(k+j,j=1..n), k=2..n), n=0..8); # Zerinvary Lajos, Jun 01 2007

Formula

a(n) = (n+2)!*(n+3)!*...*(2n)!/(2!*3!*...*n!) = A000178(2n)/(A000178(n)*A000178(n+1)) = A079478(n)/A000142(n+1).
a(n) ~ A * 2^(2*n^2 + 2*n - 7/12) * n^(n^2 - n - 23/12) / (Pi * exp(3*n^2/2 - n + 1/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Jul 10 2015

A192668 Floor-Sqrt transform of superfactorials (A000178).

Original entry on oeis.org

1, 1, 1, 3, 16, 185, 4988, 354134, 71109667, 42836123450, 81600285441318, 515548511098996334, 11283348939893661586501, 890385701589932763452676123, 262895016275494870674135139820802, 300629890583706167610723324054426034948
Offset: 0

Views

Author

Emanuele Munarini, Jul 07 2011

Keywords

Links

Crossrefs

Cf. A192660-A192665, A192668-A192685.

Programs

  • Mathematica
    Table[Floor[Sqrt[Product[k!,{k,0,n}]]],{n,0,18}]
  • Maxima
    makelist(floor(sqrt(product(k!,k,0,n))),n,0,12);
  • PARI
    a(n) = sqrtint(prod(k=0, n, k!)); \\ Michel Marcus, Apr 08 2021

Formula

a(n) = floor(sqrt(Product_{k=0..n} k!)).

Extensions

Definition corrected by Georg Fischer, Apr 08 2021

A193478 G.f. A(x) satisfies: 1/(1-x) = Sum_{n>=0} A(x)^n/sf(n), where A(x) = Sum_{n>=1} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).

Original entry on oeis.org

1, 1, 5, 95, 9959, 6270119, 28519938719, 1045680030158399, 349874346597600908159, 1178635679994967168072291199, 44013684086180240167822552866892799, 19826711369458419136710617483545735797772799, 116690731684609551482643899854886684445978037938815999
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2011

Keywords

Examples

			A(x) = x + x^2/(1!*2!) + 5*x^3/(1!*2!*3!) + 95*x^4/(1!*2!*3!*4!) + 9959*x^5/ (1!*2!*3!*4!*5!) + 6270119*x^6/(1!*2!*3!*4!*5!*6!) +...+ a(n)*x^n/sf(n) +...
where
1/(1-x) = 1 + A(x) + A(x)^2/(1!*2!) + A(x)^3/(1!*2!*3!) + A(x)^4/(1!*2!*3!*4!) + A(x)^5/(1!*2!*3!*4!*5!) + A(x)^6/(1!*2!*3!*4!*5!*6!) +...+  A(x)^n/sf(n) +...
and sf(n) = 0!*1!*2!*3!*...*(n-1)!*n!.

Crossrefs

Cf. A000178, A193479, A193440.

Programs

  • PARI
    {a(n)=local(A=sum(m=1,n-1,a(m)*x^m/prod(k=0,m,k!))+O(x^(n+2)));
prod(k=0,n,k!)*polcoeff(1/(1-x)-sum(m=0,n,A^m/prod(k=0,m,k!)),n)}

A193479 G.f. A(x) satisfies: 1+x = Sum_{n>=0} A(x)^n/sf(n), where A(x) = Sum_{n>=1} a(n)*x^n/sf(n), and sf(n) = Product_{k=0..n} k! is the superfactorial of n (A000178).

Original entry on oeis.org

1, -1, 5, -121, 16199, -13857481, 86631572159, -4470597876144961, 2126428452257713430399, -10305779379533133607589385601, 557802385738943120790269629003660799, -366846102335019802908345392106358106684889601, 3169417347948517943104654704100947667168800468999705599
Offset: 1

Views

Author

Paul D. Hanna, Jul 27 2011

Keywords

Examples

			A(x) = x - x^2/(1!*2!) + 5*x^3/(1!*2!*3!) - 121*x^4/(1!*2!*3!*4!) + 16199*x^5/(1!*2!*3!*4!*5!) - 13857481*x^6/(1!*2!*3!*4!*5!*6!) +...+ a(n)*x^n/sf(n) +...
where
1+x = 1 + A(x) + A(x)^2/(1!*2!) + A(x)^3/(1!*2!*3!) + A(x)^4/(1!*2!*3!*4!) + A(x)^5/(1!*2!*3!*4!*5!) + A(x)^6/(1!*2!*3!*4!*5!*6!) +...+  A(x)^n/sf(n) +...
and sf(n) = 0!*1!*2!*3!*...*(n-1)!*n!.

Crossrefs

Cf. A000178, A193478, A193440.

Programs

  • PARI
    {a(n)=local(A=sum(m=1,n-1,a(m)*x^m/prod(k=0,m,k!))+O(x^(n+2)));
prod(k=0,n,k!)*polcoeff(1+x-sum(m=0,n,A^m/prod(k=0,m,k!)),n)}

A198892 E.g.f.: 1/[ Sum_{n>=0} (-x)^(n*(n+1)/2) / A000178(n) ] where A000178(n) = Product_{k=1..n} k!.

Original entry on oeis.org

1, 1, 2, 9, 48, 300, 2280, 20580, 211680, 2434320, 31134600, 438807600, 6744276000, 112237725600, 2011760150400, 38639999197800, 791610365145600, 17230493212732800, 397111119429024000, 9660782144094681600, 247393077222459168000, 6651976858409613931200
Offset: 0

Views

Author

Paul D. Hanna, Oct 30 2011

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 48*x^4/4! + 300*x^5/5! +...
where
1/A(x) = 1 - x/1! - x^3/(1!*2!) + x^6/(1!*2!*3!) + x^10/(1!*2!*3!*4!) - x^15/(1!*2!*3!*4!*5!) - x^21/(1!*2!*3!*4!*5!*6!) ++--...
1/A(x) = 1 - x - x^3/2 + x^6/12 + x^10/288 - x^15/34560 - x^21/24883200 +...

Crossrefs

Cf. A198891.

Programs

  • PARI
    {a(n) = my(A=1/sum(m=0,sqrtint(2*n+1), (-x)^(m*(m+1)/2) / prod(k=1,m,k!)+x*O(x^n))); n!*polcoeff(A,n)}
for(n=0,25,print1(a(n),", "))
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