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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A236407 a(n) = 2*Sum_{k=0..n-1} C(n-1,k)*C(n+k,k) + n.

Original entry on oeis.org

0, 3, 10, 41, 196, 1007, 5342, 28821, 157192, 864155, 4780018, 26572097, 148321356, 830764807, 4666890950, 26283115053, 148348809232, 838944980531, 4752575891162, 26964373486425, 153196621856212, 871460014012703, 4962895187697070, 28292329581548741
Offset: 0

Views

Author

N. J. A. Sloane, Jan 31 2014

Keywords

Links

Crossrefs

Cf. A002003.

Programs

  • Mathematica
    Table[2*Sum[Binomial[n-1,k]*Binomial[n+k,k],{k,0,n-1}]+n,{n,0,20}] (* Vaclav Kotesovec, Feb 14 2014 *)
Flatten[{0,Table[n+2*Hypergeometric2F1[1-n,1+n,1,-1],{n,1,20}]}] (* Vaclav Kotesovec, Feb 14 2014 *)
  • PARI
    for(n=0,25, print1(n + 2*sum(k=0,n-1, binomial(n-1,k) * binomial(n+k,k)), ", ")) \\ G. C. Greubel, Jun 01 2017

Formula

a(n) = A002003(n) + n.
Conjecture: n*(n-3)*a(n) -4*(2*n-1)*(n-3)*a(n-1) +2*(7*n^2-28*n+20)*a(n-2) -4*(n-1)*(2*n-7)*a(n-3) +(n-1)*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 01 2014
Recurrence: (n-2)*n*(2*n^2 - 8*n + 7)*a(n) = (14*n^4 - 88*n^3 + 189*n^2 - 158*n + 39)*a(n-1) - (14*n^4 - 80*n^3 + 153*n^2 - 112*n + 24)*a(n-2) + (n-3)*(n-1)*(2*n^2 - 4*n + 1)*a(n-3). - Vaclav Kotesovec, Feb 14 2014
a(n) ~ 2^(-3/4) * (3+2*sqrt(2))^n / sqrt(Pi*n). - Vaclav Kotesovec, Feb 14 2014

A344563 T(n, k) = binomial(n - 1, k - 1) * binomial(n, k) * 2^k, T(0, 0) = 1. Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 2, 0, 4, 4, 0, 6, 24, 8, 0, 8, 72, 96, 16, 0, 10, 160, 480, 320, 32, 0, 12, 300, 1600, 2400, 960, 64, 0, 14, 504, 4200, 11200, 10080, 2688, 128, 0, 16, 784, 9408, 39200, 62720, 37632, 7168, 256, 0, 18, 1152, 18816, 112896, 282240, 301056, 129024, 18432, 512
Offset: 0

Views

Author

Peter Luschny, May 30 2021

Keywords

Examples

			[0] 1;
[1] 0,  2;
[2] 0,  4,    4;
[3] 0,  6,   24,     8;
[4] 0,  8,   72,    96,     16;
[5] 0, 10,  160,   480,    320,     32;
[6] 0, 12,  300,  1600,   2400,    960,     64;
[7] 0, 14,  504,  4200,  11200,  10080,   2688,    128;
[8] 0, 16,  784,  9408,  39200,  62720,  37632,   7168,   256;
[9] 0, 18, 1152, 18816, 112896, 282240, 301056, 129024, 18432, 512.

Links

Crossrefs

Row sums are A002003 with a(0) = 1, cf. also A047781.
The coefficients of the associated polynomials are in A103371.

Programs

  • Maple
    aRow := n -> seq(binomial(n-1, k-1)*binomial(n,k)*2^k, k=0..n):
seq(print(aRow(n)), n=0..9);
  • Mathematica
    T[n_, k_] := Binomial[n-1, k-1] * Binomial[n, k] * 2^k;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
  • Python
    from math import comb
def T(n, k):
    return comb(n-1, k-1)*comb(n, k)*2**k if k > 0 else k**n
print([T(n, k) for n in range(10) for k in range(n+1)]) # Michael S. Branicky, May 30 2021

A371698 Number of partial order-preserving or -reversing transformations of a chain of length n.

Original entry on oeis.org

2, 9, 54, 323, 1848, 10293, 56738, 312327, 1723692, 9549785, 53121654, 296593547, 1661423104, 9333552509, 52565738570, 296696569871, 1677887732820, 9505147063713, 53928737011358, 306393222740883, 1742919983985192, 9925790283119429, 56584658970159474, 322879453747840023
Offset: 1

Views

Author

James Mitchell, Apr 03 2024

Keywords

Links

Crossrefs

Cf. A002003.

Programs

  • GAP
    List([1..40], n -> 4 * Sum([0 .. n - 1], k ->  Binomial(n - 1, k) * Binomial(n + k, k)) - (1 + n * (2 ^ n - 1)));
  • PARI
    a(n) = 4 * sum(k=0, n-1, binomial(n -1, k)*binomial(n + k, k)) - (1 + n * (2 ^ n - 1)); \\ Michel Marcus, Apr 03 2024

Formula

a(n) = 4*Sum_{k=0..n-1} binomial(n-1, k)*binomial(n+k, k) - (1 + n*(2 ^ n - 1)).
a(n) = 2*A002003(n) - (1 + n*(2^n - 1)).
Previous Showing 21-23 of 23 results.

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

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