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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A308876 Expansion of e.g.f. exp(x)*(1 - x)/(1 - 2*x).

Original entry on oeis.org

1, 2, 7, 40, 317, 3166, 37987, 531812, 8508985, 153161722, 3063234431, 67391157472, 1617387779317, 42052082262230, 1177458303342427, 35323749100272796, 1130359971208729457, 38432239021096801522, 1383560604759484854775, 52575302980860424481432
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 29 2019

Keywords

Comments

Binomial transform of A002866.

Links

Crossrefs

Cf. A002866, A010844, A011782, A067273, A161936, A271705, A294466, A327606.
Row sums of A326659.

Programs

  • Maple
    a:= n-> n! * add(ceil(2^(n-k-1))/k!, k=0..n):
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 12 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[Exp[x] (1 - x)/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
Table[1 + Sum[Binomial[n,k] 2^(k - 1) k!, {k, 1, n}], {n, 0, 19}]

Formula

a(n) = 1 + Sum_{k=1..n} binomial(n,k) * 2^(k-1) * k!.
a(n) = A010844(n) - A067273(n).
a(n) ~ n! * 2^(n-1) * exp(1/2). - Vaclav Kotesovec, Jun 29 2019
a(n) = Sum_{k=0..n} k! * A271705(n,k). - Alois P. Heinz, Sep 12 2019

A326659 T(n,k) = [0=0]; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 1, 15, 18, 6, 1, 64, 132, 96, 24, 1, 325, 980, 1140, 600, 120, 1, 1956, 7830, 12720, 10440, 4320, 720, 1, 13699, 68502, 143850, 162120, 103320, 35280, 5040, 1, 109600, 657608, 1698816, 2447760, 2123520, 1108800, 322560, 40320
Offset: 0

Views

Author

Alois P. Heinz, Sep 12 2019

Keywords

Comments

[] is an Iverson bracket.

Examples

			Triangle T(n,k) begins:
  1;
  1,     1;
  1,     4,     2;
  1,    15,    18,      6;
  1,    64,   132,     96,     24;
  1,   325,   980,   1140,    600,    120;
  1,  1956,  7830,  12720,  10440,   4320,   720;
  1, 13699, 68502, 143850, 162120, 103320, 35280, 5040;
  ...

Links

Crossrefs

Columns k=0-2 give: A000012, A007526, 2*A134432(n-1).
Main diagonal gives A000142.
Row sums give A308876.
Cf. A000522, A073474, A093964, A143409, A196347, A271705, A327606.

Programs

  • Maple
    T:= proc(n, k) option remember;
      `if`(0=0, 1, 0)
    end:
seq(seq(T(n, k), k=0..n), n=0..10);
  • Mathematica
    T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = Boole[0 < k <= n]*n*(T[n-1, k-1] + T[n-1, k]) + Boole[k == 0 && n >= 0];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)

Formula

E.g.f. of column k: exp(x)*(x/(1-x))^k.
T(n,k) = k! * A271705(n,k).
T(n,k) = n * A073474(n-1,k-1) for n,k >= 1.
T(n,1) = n * A000522(n-1) for n >= 1.
T(n,2) = n * A093964(n-1) for n >= 1.
Sum_{k=1..n} k * T(n,k) = A327606(n).
Showing 1-2 of 2 results.

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

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