A095989 -id:A095989 - cp's OEIS frontend

A109062 Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n.

1, 1, 1, 1, 4, 3, 1, 11, 23, 13, 1, 26, 112, 158, 71, 1, 57, 446, 1170, 1241, 461, 1, 120, 1593, 6880, 12871, 10912, 3447, 1, 247, 5337, 35503, 103887, 150413, 106031, 29093, 1, 502, 17190, 168982, 724148, 1589266, 1872286, 1128218, 273343, 1, 1013, 54008

Offset: 1

Mike Zabrocki, Aug 24 2005

Also the number of free generators and primitives of the quasi-symmetric functions in non-commuting variables. - Mike Zabrocki, Aug 06 2006
Triangle given by [1,0,2,0,3,0,4,0,5,...] DELTA [1,2,2,3,3,4,4,5,5,6,6,7,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 01 2007
Apparently, the alternating sums vanish for n > 1. - F. Chapoton, Sep 05 2023

			Atomic set compositions a(1,1)=1: [{1}]; a(2,1)=1, a(2,2)=1: [{12}], [{2},{1}]; a(3,1)=1, a(3,2)=4, a(3,3)=3: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}].
Triangle begins:
  1;
  1,  1;
  1,  4,   3;
  1, 11,  23,  13;
  1, 26, 112, 158, 71;
  ...

N. Bergeron and M. Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree, arXiv:math/0509265 [math.CO], 2005.
Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 7.

Row sums are equal to A095989, a(n,n) = A003319, a(n,2) = A000295.

Cf. A095989, A059438, A074664, A087903, A008277, A019538.

f:=(n,k)->coeff(coeff(series(1-1/(1+add(add(q^m*t^i*
    Stirling2(m,i)*i!,i=1..m),m=1..n)),q,n+1),q,n),t,k):
seq(seq(f(n,k), k=1..n), n=1..10);

G.f.: 1-1/(1+Sum_{n>=1} Sum_{k=1..n} q^n*t^k*Stirling2(n,k)*k!).

A163204 Triangle read by rows, A095989 convolved with A000670.

Original entry on oeis.org

1, 1, 2, 3, 2, 8, 13, 6, 8, 48, 75, 26, 24, 48, 368, 541, 150, 104, 144, 368, 3376, 4683, 1082, 600, 624, 1104, 3376, 35824, 47293, 9366, 4328, 3600, 4784, 10128, 35824, 430512, 545835, 94586, 37464, 25968, 27600, 43888, 107472, 430512, 5773936, 7087261, 1091670, 378344, 224784, 199088, 253200, 465712, 1291536, 5773936, 85482032

Offset: 1

Gary W. Adamson, Jul 23 2009

Row sums = A000670 starting with offset 1: (1, 3, 13, 75, 541, 4683,...).
Left border = A000670, right border = A095989.
Second column: A076726. - Michel Marcus, Mar 31 2016

			First few rows of the triangle are:
1;
1, 2;
3, 2, 8;
13, 6, 8, 48;
75, 26, 24, 48, 368;
541, 150, 104, 144, 368, 3376;
4683, 1082, 600, 624, 1104, 3376, 35824;
47293, 9366, 4328, 3600, 4784, 10128, 35824, 430512;
...

G. C. Greubel, Table of n, a(n) for n = 1..1225

Cf. A000670, A076726, A095989.

max = 10; Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k - i - r)!), {i, 0, k - r}], {k, r, n}]; Fubini[0, 1] = 1; A000670 = Table[ Fubini[n, 1], {n, 0, max}]; s = 1 - 1/Sum[Fubini[k, 1] q^k, {k, 0, max}] + O[q]^max; A095989 = CoefficientList[s/q, q]; row[n_] := A095989[[1 ;; n]]*Reverse[A000670[[1 ;; n]]]; Table[row[n], {n, 1, max-1}] // Flatten (* Jean-François Alcover, Mar 31 2016 *)

Descending antidiagonals of a multiplication table formed by convolving A095989 with A000670, where A095989 is the INVERTi transform of A000670 starting (1, 3, 13, 75,...).

a(23) corrected by Jean-François Alcover, Mar 31 2016

Terms a(37) onward added by G. C. Greubel, Dec 10 2016

A095993 Inverse Euler transform of the ordered Bell numbers A000670.

Original entry on oeis.org

1, 1, 2, 10, 59, 446, 3965, 41098, 484090, 6390488, 93419519, 1498268466, 26159936547, 494036061550, 10035451706821, 218207845446062, 5057251219268460, 124462048466812950, 3241773988588098756, 89093816361187396674, 2576652694087142999421

Offset: 0

Mike Zabrocki, Jul 18 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..423

Cf. A000670, A085686, A095989.

read transforms; A000670 := proc(n) option remember; local k; if n <=1 then 1 else add(binomial(n,k)*A000670(n-k),k=1..n); fi; end; [seq(A000670(i),i=1..30)]; EULERi(%);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000670):
seq(a(n), n = 0..22); # Peter Luschny, Nov 21 2022

max = 25; b[0] = 1; b[n_] := b[n] = Sum[Binomial[n, k]*b[n-k], {k, 1, n}]; bb = Array[b, max]; s = {}; For[i=1, i <= max, i++, AppendTo[s, i*bb[[i]] - Sum[s[[d]]*bb[[i-d]], {d, i-1}]]]; a[0] = 1; a[n_] := Sum[If[Divisible[ n, d], MoebiusMu[n/d], 0]*s[[d]], {d, 1, n}]/n; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Feb 25 2017 *)

Product(1/(1-q^n)^(a(n)), n >=1) = sum(A000670(k)*q^k, k>=0).

a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Oct 09 2019

a(0)=1 inserted by Alois P. Heinz, Feb 20 2017

cp's OEIS Frontend

A109062 Triangle read by rows: number of atomic set compositions of size n and length k (see description in A095989) 1 <= k <= n.

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A163204 Triangle read by rows, A095989 convolved with A000670.

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A095993 Inverse Euler transform of the ordered Bell numbers A000670.

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