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 1-7 of 7 results.

A089041 Inverse binomial transform of squares of factorial numbers.

Original entry on oeis.org

1, 0, 3, 26, 453, 11844, 439975, 22056222, 1436236809, 117923229512, 11921584264011, 1455483251191650, 211163237294447053, 35913642489947449356, 7077505637217289437423, 1599980633296779087784934, 411293643476907595937924625, 119299057697083019137937718672
Offset: 0

Views

Author

Vladeta Jovovic, Dec 03 2003

Keywords

Comments

a(n) enumerates (ordered) lists of n two-tuples such that all numbers from 1 to n appear as the first as well as the second tuple entry and the j-th list member is not the tuple (j,j), for every j=1,..,n. Called coincidence-free 2-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.

Examples

			2-tuple combinatorics: a(1)=0 because the only list of 2-tuples with numbers 1 is [(1,1)] and this is a coincidence for j=1.
2-tuple combinatorics: the a(2)=3 coincidence free 2-tuple lists of length n=2 are [(1,2),(2,1)], [(2,1),(1,2)] and [(2,2),(1,1)]. The list [(1,1),(2,2)] has two coincidences (j=1 and j=2).

References

  • Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=2.

Links

Crossrefs

Cf. A001044, A046662 (binomial transform of squares of factorial numbers).
(-1)^n times the polynomials in A099599 evaluated at -1.

Programs

  • Maple
    a:= proc(n) option remember;
       `if`(n<2, 1-n, n^2*a(n-1)+n*(n-1)*a(n-2)+(-1)^n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 21 2013
  • Mathematica
    Table[n!Sum[(-1)^k(n-k)!/k!,{k,0,n}],{n,0,15}] (* Geoffrey Critzer, Jun 17 2013 *)

Formula

G.f.: hypergeom([1, 1, 1], [], x/(1+x))/(1+x).
E.g.f.: exp(-x)* hypergeom([1, 1], [], x).
a(n) = n^2*a(n-1) + n*(n-1)*a(n-2) + (-1)^n. - Vladeta Jovovic, Jul 15 2004
a(n) = Sum_{j=0..n} ((-1)^(n-j))*binomial(n,j)*(j!)^2. See the Charalambides reference a(n)=B_{n,2}.
a(n) = (n-1)*(n+1)*a(n-1) + (n-1)*(2*n-1)*a(n-2) + (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Aug 13 2013
a(n) ~ 2*Pi*exp(-2*n)*n^(2*n+1). - Vaclav Kotesovec, Aug 13 2013
G.f.: Sum_{k>=0} (k!)^2*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019

Extensions

Charalambides reference and comments with combinatorial examples from Wolfdieter Lang, Jan 21 2008

A343898 a(n) = Sum_{k=0..n} (k!)^3 * binomial(n,k).

Original entry on oeis.org

1, 2, 11, 244, 14741, 1799366, 383827807, 130673579576, 66583061972009, 48379301165408266, 48265538214413425331, 64129741094923528310012, 110669722298686436099306941, 242891356723607474283206170574, 665950191893557715599111566813191, 2246102991406652396042587363523672896
Offset: 0

Views

Author

Seiichi Manyama, May 03 2021

Keywords

Comments

Binomial transform of (n!)^3.

Links

Crossrefs

Cf. A000522, A046662, A343899, A343900.

Programs

  • Mathematica
    a[n_] := Sum[(k!)^3 * Binomial[n, k], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, May 05 2021 *)
  • PARI
    a(n) = sum(k=0, n, k!^3*binomial(n, k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!^3*x^k/(1-x)^(k+1)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, k!^2*x^k)))

Formula

G.f.: Sum_{k>=0} (k!)^3 * x^k/(1 - x)^(k+1).
E.g.f.: exp(x) * Sum_{k>=0} (k!)^2 * x^k.
a(n) ~ (n!)^3. - Vaclav Kotesovec, May 03 2021

A343899 a(n) = Sum_{k=0..n} (k!)^k * binomial(n,k).

Original entry on oeis.org

1, 2, 7, 232, 332669, 24884861086, 139314218808181027, 82606412229102532926819812, 6984964247802365417561163907914436537, 109110688415634181158572146813823590758078301022074, 395940866122426284350759726810156652343313286283891529199276099071
Offset: 0

Views

Author

Seiichi Manyama, May 03 2021

Keywords

Comments

Binomial transform of (n!)^n.

Links

Crossrefs

Cf. A000522, A036740, A046662, A086331, A343898, A343900, A343928.

Programs

  • Mathematica
    a[n_] := Sum[(k!)^k * Binomial[n, k], {k, 0, n}]; Array[a, 11, 0] (* Amiram Eldar, May 05 2021 *)
  • PARI
    a(n) = sum(k=0, n, k!^k*binomial(n, k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k!*x)^k/(1-x)^(k+1)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, k!^(k-1)*x^k)))

Formula

G.f.: Sum_{k>=0} (k! * x)^k/(1 - x)^(k+1).
E.g.f.: exp(x) * Sum_{k>=0} (k!)^(k-1) * x^k.

A343900 a(n) = Sum_{k=0..n} (k!)^(k+1) * binomial(n,k).

Original entry on oeis.org

1, 2, 11, 1324, 7967861, 2986023826166, 100306147958903465407, 416336313421816733159702737376, 281633758448076539969292901914477101456489, 39594086612245054028213574779019294652734771094507399786
Offset: 0

Views

Author

Seiichi Manyama, May 03 2021

Keywords

Comments

Binomial transform of (n!)^(n+1).

Links

Crossrefs

Cf. A000522, A036740, A046662, A091868, A343898, A343899, A343929.

Programs

  • Mathematica
    a[n_] := Sum[(k!)^(k+1) * Binomial[n, k], {k, 0, n}]; Array[a, 10, 0] (* Amiram Eldar, May 05 2021 *)
  • PARI
    a(n) = sum(k=0, n, k!^(k+1)*binomial(n, k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!^(k+1)*x^k/(1-x)^(k+1)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x)*sum(k=0, N, (k!*x)^k)))

Formula

G.f.: Sum_{k>=0} (k!)^(k+1) * x^k/(1 - x)^(k+1).
E.g.f.: exp(x) * Sum_{k>=0} (k! * x)^k.

A343928 a(n) = Sum_{k=0..n} (k!)^n * binomial(n,k).

Original entry on oeis.org

1, 2, 7, 244, 337061, 24923091206, 139331988275478727, 82607113404338664216300296, 6984967577834038055008791270166057993, 109110690950275218023122492287310115968068596613130, 395940866518366059877297056617763923418318903997411043997258716171
Offset: 0

Views

Author

Seiichi Manyama, May 04 2021

Keywords

Links

Crossrefs

Cf. A000522, A046662, A072034, A336247, A339311, A343898, A343899, A343929.

Programs

  • Mathematica
    a[n_] := Sum[(k!)^n * Binomial[n, k], {k, 0, n} ]; Array[a, 11, 0] (* Amiram Eldar, May 04 2021 *)
  • PARI
    a(n) = sum(k=0, n, k!^n*binomial(n, k));

Formula

a(n) = [x^n] Sum_{k>=0} (k!)^n * x^k/(1 - x)^(k+1).
a(n) = n! * [x^n] exp(x) * Sum_{k>=0} (k!)^(n-1) * x^k.

A343929 a(n) = Sum_{k=0..n} (k!)^(n+1) * binomial(n,k).

Original entry on oeis.org

1, 2, 11, 1348, 7993925, 2986939982086, 100308280020162672007, 416336818263472141683094788104, 281633775231427434285800695714399092181001, 39594086714441777969538839399390619086007952991080833034
Offset: 0

Views

Author

Seiichi Manyama, May 04 2021

Keywords

Links

Crossrefs

Cf. A046662, A343898, A343900, A343928.

Programs

  • Mathematica
    a[n_] := Sum[(k!)^(n+1) * Binomial[n, k], {k, 0, n} ]; Array[a, 10, 0] (* Amiram Eldar, May 04 2021 *)
  • PARI
    a(n) = sum(k=0, n, k!^(n+1)*binomial(n, k));

Formula

a(n) = [x^n] Sum_{k>=0} (k!)^(n+1) * x^k/(1 - x)^(k+1).
a(n) = n! * [x^n] exp(x) * Sum_{k>=0} (k!)^n * x^k.

A371767 Triangle read by rows: T(n, k) = (k! * n!)/(n - k)!.

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 12, 36, 1, 4, 24, 144, 576, 1, 5, 40, 360, 2880, 14400, 1, 6, 60, 720, 8640, 86400, 518400, 1, 7, 84, 1260, 20160, 302400, 3628800, 25401600, 1, 8, 112, 2016, 40320, 806400, 14515200, 203212800, 1625702400
Offset: 0

Views

Author

Peter Luschny, Apr 14 2024

Keywords

Examples

			Triangle starts:
  [0] 1;
  [1] 1, 1;
  [2] 1, 2,  4;
  [3] 1, 3, 12,   36;
  [4] 1, 4, 24,  144,   576;
  [5] 1, 5, 40,  360,  2880,  14400;
  [6] 1, 6, 60,  720,  8640,  86400,  518400;
  [7] 1, 7, 84, 1260, 20160, 302400, 3628800, 25401600;

Crossrefs

Cf. A000142, A001044 (main diagonal), A010790 (subdiagonal), A046662 (row sums), A089041 (alternating row sums), A010050 (central terms).

Programs

  • Maple
    T := (n, k) -> (k! * n!)/(n - k)!:
for n from 0 to 6 do seq(T(n, k), k = 0..n) od;
Showing 1-7 of 7 results.

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