A057837 -id:A057837 - cp's OEIS frontend

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

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).

A347435 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6) ).

Original entry on oeis.org

1, 0, 0, 0, 1, 6, 22, 64, 198, 1138, 10004, 83920, 617993, 4226028, 30103686, 251883012, 2490287821, 26456763078, 281404300348, 2966101610920, 31877462564554, 362624252399566, 4437794875670072, 57612897938229380, 773900876490016325, 10599854900351622752

Offset: 0

Ilya Gutkovskiy, Sep 02 2021

nonn

Exponential transform of A002663.

Cf. A002663, A055882, A057837, A143405, A347432, A347434.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*(2^j-j^3/6-5*j/6-1), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2021

nmax = 25; CoefficientList[Series[Exp[Exp[x] (Exp[x] - 1 - x - x^2/2 - x^3/6)], {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^2 + 5)/6) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A002663(k) * a(n-k).

A365894 Expansion of e.g.f. exp( Sum_{k>=0} x^(3k+4) / (3k+4)! ).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 35, 0, 1, 330, 5775, 1, 2717, 225225, 2627626, 21828, 6782490, 290990701, 2546343368, 190030590, 22939766851, 644182060203, 4514461227804, 1607617027501, 109664100094160, 2261215037103165, 13296854061626851, 15998661864449331

Offset: 0

Seiichi Manyama, Sep 22 2023

nonn

Cf. A057837, A365895.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sum(k=0, N\3, x^(3*k+4)/(3*k+4)!))))

a(0)=1; a(n) = Sum_{k=0..floor((n-4)/3)} binomial(n-1,3*k+3) * a(n-3*k-4).

cp's OEIS Frontend

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

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A347435 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6) ).

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A365894 Expansion of e.g.f. exp( Sum_{k>=0} x^(3k+4) / (3k+4)! ).

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cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Maxima

Formula

A347435 E.g.f.: exp( exp(x) * (exp(x) - 1 - x - x^2 / 2 - x^3 / 6) ).

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula

A365894 Expansion of e.g.f. exp( Sum_{k>=0} x^(3*k+4) / (3*k+4)! ).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A365894 Expansion of e.g.f. exp( Sum_{k>=0} x^(3k+4) / (3k+4)! ).