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.

A096131 Sum of the terms of the n-th row of triangle pertaining to A096130.

Original entry on oeis.org

1, 7, 105, 2386, 71890, 2695652, 120907185, 6312179764, 375971507406, 25160695768715, 1869031937691061, 152603843369288819, 13584174777196666630, 1309317592648179024666, 135850890740575408906465
Offset: 1

Views

Author

Amarnath Murthy, Jul 04 2004

Keywords

Comments

The product of the terms of the n-th row is given by A034841.
Collection of partial binary matrices: 1 to n rows of length n and a total of n entries set to one in each partial matrix. - Olivier Gérard, Aug 08 2016

Examples

			From _Seiichi Manyama_, Aug 18 2018: (Start)
a(1) = (1/1!) * (1) = 1.
a(2) = (1/2!) * (1*2 + 3*4) = 7.
a(3) = (1/3!) * (1*2*3 + 4*5*6 + 7*8*9) = 105.
a(4) = (1/4!) * (1*2*3*4 + 5*6*7*8 + 9*10*11*12 + 13*14*15*16) = 2386. (End)

Links

Crossrefs

Cf. A014062, A096130, A034841, A007318, A226391, A167009, A167008, A167010, A072034, A086331, A349470.

Programs

  • GAP
    List(List([1..20],n->List([1..n],k->Binomial(k*n,n))),Sum); # Muniru A Asiru, Aug 12 2018
  • Maple
    A096130 := proc(n,k) binomial(k*n,n) ; end: A096131 := proc(n) local k; add( A096130(n,k),k=1..n) ; end: for n from 1 to 18 do printf("%d, ",A096131(n)) ; od ; # R. J. Mathar, Apr 30 2007
seq(add((binomial(n*k,n)), k=0..n), n=1..15); # Zerinvary Lajos, Sep 16 2007
  • Mathematica
    Table[Sum[Binomial[k*n, n], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jun 06 2013 *)
  • PARI
    a(n) = sum(k=1, n, binomial(k*n, n)); \\ Michel Marcus, Aug 20 2018

Formula

a(n) = Sum_{k=1..n} binomial(k*n, n). - Reinhard Zumkeller, Jan 09 2005
a(n) = (1/n!) * Sum_{j=1..n} Product_{i=n*(j-1)+1..n*j} i. - Reinhard Zumkeller, Jan 09 2005 [corrected by Seiichi Manyama, Aug 17 2018]
a(n) ~ exp(1)/(exp(1)-1) * binomial(n^2,n). - Vaclav Kotesovec, Jun 06 2013

Extensions

More terms from R. J. Mathar, Apr 30 2007
Edited by N. J. A. Sloane, Sep 06 2008 at the suggestion of R. J. Mathar

A349471 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,k).

Original entry on oeis.org

1, 0, 5, 71, 1625, 48699, 1815157, 80960200, 4205895521, 249447427145, 16631893722851, 1231521399730489, 100270564101729529, 8903719880410535595, 856322102196977446955, 88677383473792696758599, 9837660365763014667911553, 1163993530309417488368300653
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2021

Keywords

Crossrefs

Cf. A226391, A349470.

Programs

  • Mathematica
    a[n_] := Sum[(-1)^(n - k) * Binomial[k*n, k], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Nov 19 2021 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*binomial(k*n, k));

Formula

a(n) ~ exp(n - 1/2) * n^(n - 1/2) / sqrt(2*Pi). - Vaclav Kotesovec, Nov 20 2021

A349480 a(n) = Sum_{j=0..n} (-1)^(n-j) * Product_{k=(j-1)*n+1..j*n} k.

Original entry on oeis.org

1, 1, 10, 390, 33456, 4845360, 1059099840, 325460948400, 133697543616000, 70733019878196480, 46831083260349024000, 37927830201482962540800, 36883442511877368877747200, 42409212946187708288828160000
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2021

Keywords

Examples

			a(2) = -1*2 + 3*4 = 10.
a(3) = 1*2*3 - 4*5*6 + 7*8*9 = 390.
a(4) = -1*2*3*4 + 5*6*7*8 - 9*10*11*12 + 13*14*15*16 = 33456.

Links

Crossrefs

Cf. A336513, A349470.

Programs

  • Mathematica
    a[n_] := n! * Sum[(-1)^(n - k) * Binomial[k*n, n], {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, Nov 19 2021 *)
  • PARI
    a(n) = sum(j=0, n, (-1)^(n-j)*prod(k=(j-1)*n+1, j*n, k));

Formula

a(n) = n! * A349470(n) = n! * Sum_{k=0..n} (-1)^(n-k) * binomial(k*n,n).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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