A107107 -id:A107107 - cp's OEIS frontend

A107106 Divide A036039(n) by A036040(n).

1, 1, 1, 2, 1, 1, 6, 2, 1, 1, 1, 24, 6, 2, 2, 1, 1, 1, 120, 24, 6, 4, 6, 2, 1, 2, 1, 1, 1, 720, 120, 24, 12, 24, 6, 4, 2, 6, 2, 1, 2, 1, 1, 1, 5040, 720, 120, 48, 36, 120, 24, 12, 6, 4, 24, 6, 4, 2, 1, 6, 2, 1, 2, 1, 1, 1, 40320, 5040, 720, 240, 144, 720, 120, 48, 36, 24, 12, 8, 120, 24, 12

Offset: 1

Alford Arnold, May 12 2005

A107107 gives the row sums. - R. J. Mathar, Aug 13 2007

This array is the first one (K=1) of a family of partition number arrays called M31(1). For M31(2) see A134133 = M_3(2)/M_3.

			a(36) = 280/70 = 4.
As array: [1];[1,1];[2,1,1];[6,2,1,1,1];[24,6,2,2,1,1,1];[120,24,6,4,6,2,1,2,1,1,1];...

Jean-François Alcover after Wolfdieter Lang, Table of n, a(n) for n = 1..138
Wolfdieter Lang, First 10 rows of the array and more.

Cf. A107107.

sortAbrSteg := proc(L1,L2) local i ; if nops(L1) < nops(L2) then RETURN(true) ; elif nops(L2) < nops(L1) then RETURN(false) ; else for i from 1 to nops(L1) do if op(i,L1) < op(i,L2) then RETURN(false) ; fi ; od ; RETURN(true) ; fi ; end: M2overM3 := proc(L) local n,k,an,resul; n := add(i,i=L) ; resul := 1 ; for k from 1 to n do an := add(1-min(abs(j-k),1),j=L) ; resul := resul* (factorial(k-1))^an ; od ; end: A107106 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then M2overM3(op(k,prts)) ; else 0 ; fi ; end: for n from 1 to 10 do for k from 1 to combinat[numbpart](n) do a:=A107106(n,k) ; printf("%d,",a) ; od; od ; # R. J. Mathar, Aug 13 2007

aspartitions[n_] := Reverse /@ Sort[Sort /@ IntegerPartitions[n]];
A036039[n_] := n!/(Times @@ #)& /@ ((#! Range[n]^#)& /@ Function[par, Count[par, #]& /@ Range[n]] /@ aspartitions[n]);
runs[li : {__Integer}] := ((Length /@ Split[#]))&[Sort@li];
A036040[n_] := Module[{temp}, temp = Map[Reverse, Sort@(Sort /@ IntegerPartitions[n]), {1}]; Apply[Multinomial, temp, {1}]/Apply[Times, (runs /@ temp)!, {1}]];
T[n_] := A036039[n]/A036040[n];
Table[T[n], {n, 1, 10}] // Flatten
(* Jean-François Alcover, Jun 10 2023, after Wouter Meeussen in A036039 *)

a(n) = A036039(n) / A036040(n).

Corrected and extended by R. J. Mathar, Aug 13 2007

a(75) and a(76) swapped (first 36, then 24) by Wolfdieter Lang, Sep 22 2008

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

Original entry on oeis.org

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)!.

A321520 Expansion of Product_{k>=1} (1 + (k - 1)!*x^k).

Original entry on oeis.org

1, 1, 1, 3, 8, 32, 152, 882, 5964, 46644, 411564, 4056912, 44097072, 524234448, 6761911968, 94055452128, 1403047948320, 22342552398720, 378256278306240, 6783950610708480, 128480976137122560, 2562250754919421440, 53668564630447910400

Offset: 0

Ilya Gutkovskiy, Nov 12 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A107107, A265950.

b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 05 2023

nmax = 22; CoefficientList[Series[Product[(1 + (k - 1)! x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d ((d - 1)!)^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 22}]

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*d*((d - 1)!)^(k/d) ) * x^k/k).

a(n) ~ (n-1)! * (1 + 1/n + 2/n^2 + 7/n^3 + 34/n^4 + 203/n^5 + 1454/n^6 + 12321/n^7 + 121326/n^8 + 1364947/n^9 + 17301550/n^10 + ...). - Vaclav Kotesovec, Nov 13 2018

cp's OEIS Frontend

A107106 Divide A036039(n) by A036040(n).

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

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Comments

Examples

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A321520 Expansion of Product_{k>=1} (1 + (k - 1)!*x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula