A107106 -id:A107106 - cp's OEIS frontend

A144351 Lower triangular array called S1hat(1) related to partition number array A107106.

1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 24, 8, 3, 1, 1, 120, 34, 9, 3, 1, 1, 720, 156, 36, 9, 3, 1, 1, 5040, 924, 166, 37, 9, 3, 1, 1, 40320, 6144, 968, 168, 37, 9, 3, 1, 1, 362880, 48096, 6372, 978, 169, 37, 9, 3, 1, 1, 3628800, 420480, 49368, 6416, 980, 169, 37, 9, 3, 1, 1, 39916800, 4134240

Offset: 1

Wolfdieter Lang Oct 09 2008

If in the partition array M31hat(1):=A107106 entries with the same parts number m are summed one obtains this triangle of numbers S1hat(1). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.

The first three columns are A000142(n-1) (factorials), A024419 (guess), A144352.

			[1];[1,1];[2,1,1];[6,3,1,1];[24,8,3,1,1];...

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

Row sums A107107.

A134134 (S1hat(2)= S2'(2)).

a(n,m)=sum(product(|S1(1;j,1)|^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. |S1(1,n,1)|= |A008275(n,1)| = A000142(n-1) = (n-1)!.

A107107 For each partition of n, calculate (dM2/dM3) where dM2 = A036039(p) and dM3 = A036040(p); then sum over all partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 11, 37, 168, 926, 6181, 47651, 418546, 4106264, 44537519, 528408261, 6807428748, 94588717554, 1409927483625, 22437711255279, 379674820846534, 6806486383431340, 128862216628864163, 2569080120361323721, 53797824318887051264, 1180533584545138213222

Offset: 0

Alford Arnold, May 12 2005

Values for individual partitions (A107106) are factorials when all but one part of the partition has size one or two, but not usually in other cases.

			For n = 6, (120,144,90,40,90,120,15,40,45,15,1) / (1,6,15,10,15,60,15,20,45,15,1)
  equals (120,24,6,4,6,2,1,2,1,1,1) so A107107(6) = 168.

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Cf. A000142, A036039, A000110, A036040, A107106, A102189.

Cf. A077365.

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
      `if`(i>n, 0, b(n-i, i)*(i-1)!)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

nmax=20; CoefficientList[Series[Product[1/(1-(k-1)!*x^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 15 2015 *)

S(n,m):=if n=0 then 1 else if nVladimir Kruchinin, Sep 07 2014 */

For partition [], the contribution to the sum is product_i (c_i - 1)!^k_i.
G.f.: 1/Product_{m>0} (1-(m-1)!*x^m). - Vladeta Jovovic, Jul 10 2007
a(n) = S(n,1), where S(n,m) = sum(k=m..n/2, (k-1)!*S(n-k,k))+(n-1)!, S(n,n)=(n-1)!, S(0,m)=1, S(n,m)=0 for m>n. - Vladimir Kruchinin, Sep 07 2014
a(n) ~ (n-1)! * (1 + 1/n + 3/n^2 + 11/n^3 + 50/n^4 + 278/n^5 + 1861/n^6 + 14815/n^7 + 138477/n^8 + 1497775/n^9 + 18465330/n^10). - Vaclav Kotesovec, Mar 15 2015

Edited, corrected and extended by Franklin T. Adams-Watters, Nov 03 2005

More terms from Vladeta Jovovic, Jul 10 2007

A144880 Partition number array, called M31hat(3).

Original entry on oeis.org

1, 3, 1, 12, 3, 1, 60, 12, 9, 3, 1, 360, 60, 36, 12, 9, 3, 1, 2520, 360, 180, 144, 60, 36, 27, 12, 9, 3, 1, 20160, 2520, 1080, 720, 360, 180, 144, 108, 60, 36, 27, 12, 9, 3, 1, 181440, 20160, 7560, 4320, 3600, 2520, 1080, 720, 540, 432, 360, 180, 144, 108, 81, 60, 36, 27

Offset: 1

Wolfdieter Lang Oct 09 2008

Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(3;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
This is the third (K=3) member of a family of partition number arrays: A107106, A134133,...

			[1];[3,1];[12,3,1];[60,12,9,3,1];[360,60,36,12,9,3,1];...
a(4,3)= 9 = |S1(3;2,1)|^2. The relevant partition of 4 is (2^2).

W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

A144882 (row sums).

A134133 (M31hat(2) array). A144885 (M31hat(4) array).

a(n,k)= product(|S1(3;j,1)|^e(n,k,j),j=1..n) with |S1(3;n,1)|= A046089(1,n) = [1,3,12,60,...], n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n.

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A144351 Lower triangular array called S1hat(1) related to partition number array A107106.

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A107107 For each partition of n, calculate (dM2/dM3) where dM2 = A036039(p) and dM3 = A036040(p); then sum over all partitions of n.

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A144880 Partition number array, called M31hat(3).

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