A173109 -id:A173109 - cp's OEIS frontend

A087650 a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k).

1, 0, 2, 3, 12, 40, 163, 714, 3426, 17721, 98254, 580316, 3633281, 24011156, 166888166, 1216070379, 9264071768, 73600798036, 608476008123, 5224266196934, 46499892038438, 428369924118313, 4078345814329010, 40073660040755336

Offset: 0

Vladeta Jovovic, Sep 23 2003

nonn

a(n) is the number of set partitions of [n] that contain exactly one singleton block and all other blocks contain an entry > this singleton. For example, a(3)=3 counts 124/3, 134/2, 1/234 but not 123/4. - David Callan, Aug 27 2014

Partial sums are A173109. - Vladimir Reshetnikov, Oct 29 2015

			G.f. = 1 + 2*x^2 + 3*x^3 + 12*x^4 + 40*x^5 + 163*x^6 + 714*x^7 + ...

Cf. A000110, A000296, A005001, A005493, A173109.

f[n_] := Sum[ StirlingS2[n, k], {k, 1, n}]; Table[(-1)^n + Sum[(-1)^(n - k)*f[k], {k, 0, n}], {n, 0, 23}] (* Robert G. Wilson v *)
Needs["DiscreteMath`Combinatorica`"]; Table[ Sum[(-1)^(n - k)*BellB[k], {k, 0, n}], {n, 0, 23}] (* Robert G. Wilson v *)

makelist(sum((-1)^(n-k)*belln(k),k,0,n),n,0,40); /* Emanuele Munarini, Sep 27 2012 */

vector(30, n, n--; sum(k=0, n, (-1)^(n-k)*polcoeff(sum(i=0, k, prod( j=1, i, x / (1 - j*x)), x^k * O(x)), k))) \\ Altug Alkan, Oct 30 2015

def A087650_list(len): # After the formula of David Callan.
    if len == 1: return [1]
    if len == 2: return [1,0]
    R = []; A = [1]; p = -1
    for i in (0..len-1):
        A.append(A[0] - A[i])
        A[i] = A[0]
        for k in range(i, 0, -1):
            A[k-1] += A[k]
        p = -p
        R.append(A[i+1] + p)
    return R
A087650_list(24) # Peter Luschny, Aug 28 2014

E.g.f.: exp(-x)*((exp(x)-1)*exp(exp(x)-1)+1).
a(n) = (-1)^n + Bell(n) - A000296(n), with Bell(n) = A000110(n). - Wolfdieter Lang, Dec 01 2003
a(n) = A000296(n+1) + (-1)^n. - David Callan, Aug 27 2014
G.f.: 1/(1+x)/W(0), where W(k) = 1 - x/(1 - x*(k+1)/W(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 10 2014
a(0) = 1; a(n) = Sum_{k=1..n-1} binomial(n,k) * a(k-1). - Ilya Gutkovskiy, Mar 04 2021

A173108 Triangle, A000110 in every column > 0, shifted down twice.

Original entry on oeis.org

1, 1, 2, 1, 5, 1, 15, 2, 1, 52, 5, 1, 203, 15, 2, 1, 877, 52, 5, 1, 4140, 203, 15, 2, 1, 21147, 877, 52, 5, 1, 115975, 4140, 203, 15, 2, 1, 678570, 21147, 877, 52, 5, 1, 4213597, 115975, 4140, 203, 15, 2, 1, 27644437, 678570, 21147, 877, 52, 5, 1

Offset: 0

Gary W. Adamson, Feb 09 2010

Row sums = A173109: (1, 1, 3, 6, 18, 58, 221, 935, ...).

Let the triangle = M. Then lim_{n->oo} M^n = A173110: (1, 1, 3, 6, 20, 60, ...).

			First few rows of the triangle:
       1;
       1;
       2,    1;
       5,    1;
      15,    2,   1;
      52,    5,   1;
     203,   15,   2,  1;
     877,   52,   5,  1;
    4140,  203,  15,  2, 1;
   21147,  877,  52,  5, 1;
  115975, 4140, 203, 15, 2, 1;
  ...

Cf. A000110, A173109, A173110, A173111.

T[n_, k_] := BellB[n - 2 k];
Table[T[n, k], {n, 0, 10}, {k, 0, Quotient[n, 2]}] // Flatten (* Jean-François Alcover, Apr 22 2022 *)

B(n) = sum(k=0, n, stirling(n, k, 2)); \\ A000110
tabf(nn) = for (n=0, nn, for(k=0, n\2, print1(B(n-2*k), ", "));); \\ Michel Marcus, Nov 19 2022

Bell sequence in every column, for columns > 0, shifted down twice.

Keyword tabf and more terms from Michel Marcus, Nov 19 2022

A173111 Triangle read by rows, A173108 * the diagonalized variant of A173110.

Original entry on oeis.org

1, 1, 2, 1, 5, 1, 15, 2, 3, 52, 5, 3, 203, 15, 6, 6, 877, 52, 15, 6, 4140, 203, 45, 12, 20, 21147, 877, 156, 30, 20, 115975, 4140, 609, 90, 40, 60, 678570, 21147, 2631, 312, 100, 60

Offset: 0

Gary W. Adamson, Feb 09 2010

Row sums = A173110: (1, 1, 3, 6, 20, 60, 230, 950, 4420, 22230,...).

			First few rows of the triangle =
1;
1;
2, 1;
5, 1;
15, 2, 3;
52, 5, 3;
203, 15, 6, 6;
877, 52, 15, 6;
4140, 203, 45, 12, 20;
21147, 877, 156, 30, 20;
115975, 4140, 609, 90, 40, 60;
678570, 21147, 2631, 312, 100, 60;
...
Example: row 7 = termwise products of (877, 52, 5, 1) and (1, 1, 3, 6) =
(877, 52, 15, 6); where (877, 52, 5, 1) = row 7 of triangle A173108, and
(1, 1, 3, 6) = the first four terms of sequence A173109.

A173108, A173109, A173110

Let triangle A173108 = Q, and M = an infinite lower triangular matrix with A173110 as the rightmost diagonal and the rest zeros. Triangle A173111 = Q*M.

cp's OEIS Frontend

A087650 a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k).

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A173108 Triangle, A000110 in every column > 0, shifted down twice.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A173111 Triangle read by rows, A173108 * the diagonalized variant of A173110.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula