A006505 -id:A006505 - cp's OEIS frontend

A182931 Generalized Bell numbers; square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 1.

1, 1, 1, 2, 0, 1, 5, 1, 0, 1, 15, 1, 0, 0, 1, 52, 4, 1, 0, 0, 1, 203, 11, 1, 0, 0, 0, 1, 877, 41, 1, 1, 0, 0, 0, 1, 4140, 162, 11, 1, 0, 0, 0, 0, 1, 21147, 715, 36, 1, 1, 0, 0, 0, 0, 1, 115975, 3425, 92, 1, 1, 0, 0, 0, 0, 0, 1, 678570, 17722, 491, 36, 1, 1, 0, 0, 0, 0, 0, 1

Offset: 1

Peter Luschny, Apr 05 2011

			Array starts:
[k=      1       2       3       4       5]
[n=0]    1,      1,      1,      1,      1,
[n=1]    1,      0,      0,      0,      0,
[n=2]    2,      1,      0,      0,      0,
[n=3]    5,      1,      1,      0,      0,
[n=4]   15,      4,      1,      1,      0,
[n=5]   52,     11,      1,      1,      1,
[n=6]  203,     41,     11,      1,      1,
[n=7]  877,    162,     36,      1,      1,
[n=8] 4140,    715,     92,     36,      1,
   A000110,A000296,A006505,A057837,A057814, ...

E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
Peter Luschny, Set partitions

Cf. A000110, A000296, A006505, A057837, A057814, A097147.

Row sums are A097147 for n >= 1.

egf := k -> exp(exp(x)*(1-GAMMA(k,x)/GAMMA(k)));
T := (n,k) -> n!*coeff(series(egf(k),x,n+1),x,n):
seq(print(seq(T(n,k),k=1..8)),n=0..8);

egf[k_] := Exp[Exp[x] (1 - Gamma[k, x]/Gamma[k])];
T[n_, k_] := n! SeriesCoefficient[egf[k], {x, 0, n}];
Table[T[n-k+1, k], {n, 0, 11}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 13 2019 *)

E.g.f.: exp(exp(x)*(1-Gamma(k,x)/Gamma(k))); Gamma(k,x) the incomplete Gamma function.

A282988 Triangle of partitions of an n-set into boxes of size >= m.

Original entry on oeis.org

1, 2, 1, 5, 1, 1, 15, 4, 1, 1, 52, 11, 1, 1, 1, 203, 41, 11, 1, 1, 1, 877, 162, 36, 1, 1, 1, 1, 4140, 715, 92, 36, 1, 1, 1, 1, 21147, 3425, 491, 127, 1, 1, 1, 1, 1, 115975, 17722, 2557, 337, 127, 1, 1, 1, 1, 1, 678570, 98253, 11353, 793, 463, 1, 1, 1, 1, 1, 1

Offset: 1

Vladimir Kruchinin, Feb 26 2017

			Triangle T(n,m) begins:
    1;
    2,   1;
    5,   1,   1;
   15,   4,   1,   1;
   52,  11,   1,   1,   1;
  203,  41,  11,   1,   1,   1;
  877, 162,  36,   1,   1,   1,   1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A000110, A000296, A006505, A057837, A057814, A182931, A260878.

T:= proc(n, k) option remember; `if`(n=0, 1, add(
      T(n-j, k)*binomial(n-1, j-1), j=k..n))
    end:
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 28 2017

T[n_, m_] := T[n, m] = Which[Or[n == m, n == 0], 1, m == 0, 0, True, Sum[Binomial[n - 1, i + m - 1] T[n - i - m, m], {i, 0, n - m}]]; Table[T[n, m], {n, 11}, {m, n}] // Flatten (* Michael De Vlieger, Feb 26 2017 *)

T(n,m):=if n=m or n=0 then 1 else if m=0 then 0 else sum(binomial(n-1, i+m-1)*T(n-i-m,m), i, 0, n-m);

T(n,m) = Sum_{i=0..n-m} C(n-1, i+m-1)*T(n-i-m, m).

E.g.f. m column of T(n,m) is exp(exp(x)-Sum_{k=0..m} 1/k!x^k).

A347434 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2) ).

Original entry on oeis.org

1, 0, 0, 1, 5, 16, 52, 274, 1990, 14354, 99704, 730225, 6061013, 56151330, 551040830, 5597109717, 59324775741, 664973687438, 7891158217876, 98253448977890, 1273082291906394, 17124091446383666, 239333235895599762, 3476600533730954761, 52394273274018321421

Offset: 0

Ilya Gutkovskiy, Sep 02 2021

nonn

Exponential transform of A002662.

Cf. A002662, A006505, A055882, A143405, A347432, A347435.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j*(j+1)/2-1), j=1..n))
    end:
seq(a(n), n=0..24);  # Alois P. Heinz, Sep 02 2021

nmax = 24; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x - x^2/2)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] (2^k - 1 - k (k + 1)/2) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002662(k) * a(n-k).

A365893 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+3) / (5k+3)! ).

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 10, 0, 1, 280, 0, 165, 15400, 1, 30030, 1401400, 6995, 6806800, 190590401, 6506835, 1939938000, 36212380820, 4940624150, 687126039601, 9163671323015, 3761116975000, 297754623925175, 2982764271647875, 3067236941769001

Offset: 0

Seiichi Manyama, Sep 22 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..613

Cf. A006505, A009227, A307978.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\5, x^(5*k+3)/(5*k+3)!))))

a(0) = 1; a(n) = Sum_{k=0..floor((n-3)/5)} binomial(n-1,5*k+2) * a(n-5*k-3).

A352900 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into intersecting polygons.

Original entry on oeis.org

0, 0, 0, 7, 28, 79, 460, 2486, 11209, 59787, 361777, 2167635, 13577211, 91919186, 650059294, 4761980740, 36508824672, 292116858616, 2424047807182, 20847409357919, 185754041370693, 1711253802075941, 16272637412753211, 159561718074359537, 1611599794862761838, 16747401536440092104

Offset: 3

Janaka Rodrigo, Apr 07 2022

nonn

			For n=6, there are a(6) = 7 intersecting partitions of the convex hexagon. On vertices 1..6, they are the following pairs of triangles:
  {1,3,4}, {5,6,2}
  {4,5,1}, {2,3,6}
  {3,4,6}, {1,2,5}
  {2,3,5}, {1,4,6}
  {1,2,4}, {5,6,3}
  {1,6,3}, {5,4,2}
  {1,3,5}, {2,4,6}

Cf. A006505, A059022, A114997, A350248.

T2(n,k) = if (n<3, 0, if (k==1, 1, k*T2(n-1,k) + binomial(n-1,2)*T2(n-3,k-1))); \\ A059022
a5(n) = if (n<3, n==0, sum(k=1, n\3, T2(n,k))); \\ A006505
a7(n) = sum(k=ceil((n+3)/2), n, (1/(n+1) * binomial(n+1, k) * binomial(2*k-n-3, n-k)) ); \\ A114997
a(n) =  a5(n) - a7(n); \\ Michel Marcus, Apr 09 2022

a(n) = A006505(n) - A114997(n).
a(n) = Sum_{k=2..floor(n/3)} (T(n,k) - C(n+1,k)*C(n-2k-1,k-1)/(n+1)); n > 5, where T(n,k) = k*T(n-1,k) + C(n-1,2)*T(n-3,k-1); n > 5 and 1 < k <= floor(n/3), T(n,k) = 1 when k = 1.
T(n,k) = A059022(n,k) is the number of different ways to partition the set of vertices of a convex n-gon into k polygons.

More terms from Michel Marcus, Apr 09 2022

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A182931 Generalized Bell numbers; square array read by ascending antidiagonals, A(n, k) for n >= 0 and k >= 1.

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A282988 Triangle of partitions of an n-set into boxes of size >= m.

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A347434 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2) ).

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A365893 Expansion of e.g.f. exp( Sum_{k>=0} x^(5*k+3) / (5*k+3)! ).

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A352900 a(n) is the number of different ways to partition the set of vertices of a convex n-gon into intersecting polygons.

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A365893 Expansion of e.g.f. exp( Sum_{k>=0} x^(5k+3) / (5k+3)! ).