A036040 -id:A036040 - cp's OEIS frontend

A257490 Irregular triangle read by rows in which the n-th row lists multinomials (A036040) for partitions of 2n which have only even parts in Abramowitz-Stegun ordering.

1, 1, 3, 1, 15, 15, 1, 28, 35, 210, 105, 1, 45, 210, 630, 1575, 3150, 945, 1, 66, 495, 462, 1485, 13860, 5775, 13860, 51975, 51975, 10395, 1, 91, 1001, 3003, 3003, 45045, 42042, 105105, 45045, 630630, 525525, 315315, 1576575, 945945, 135135

Offset: 1

Hartmut F. W. Hoft, Apr 26 2015

The length of row n is given by A000041(n).
Each entry in this irregular triangle is the quotient of the respective entries in A257468 and A096162, which is the multinomial called M_3 in Abramowitz-Stegun.
Has the same structure as the triangles in A036036, A096162, A115621 and A257468.

			Brackets group all partitions of the same length when there is more than one partition.
n/m  1    2          3           4    5
1:   1
2:   1    3
3:   1   15         15
4:   1  [28  35]   210         105
5:   1  [45 210]  [630 1575]  3150  945
...
n = 6:  1 [66 495 462] [1485 13860 5775] [13860 51975] 51975  0395
Replacing the bracketed numbers by their sums yields the triangle of A156289.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy], pp. 831-832.

Cf. A000041, A036040, A036036, A096162, A115621, A156289, A257468.

(* triangle2574868[] and triangle096162[] are defined as functions triangle[] in the respective sequences A257468 and A096162 *)
triangle[n_] := triangle257468[n]/triangle096162[n]
a[n_] := Flatten[triangle[n]]
a[7] (* data *)

Edited by Wolfdieter Lang, May 11 2015

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

Original entry on oeis.org

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

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

A144279 Partition number array, called M32hat(-3)= 'M32(-3)/M3'= 'A143173/A036040', related to A000369(n,m)= |S2(-3;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 3, 1, 21, 3, 1, 231, 21, 9, 3, 1, 3465, 231, 63, 21, 9, 3, 1, 65835, 3465, 693, 441, 231, 63, 27, 21, 9, 3, 1, 1514205, 65835, 10395, 4851, 3465, 693, 441, 189, 231, 63, 27, 21, 9, 3, 1, 40883535, 1514205, 197505, 72765, 53361, 65835, 10395, 4851, 2079, 1323, 3465

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) =: M32hat(-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,...].
If M32hat(-3;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-3):= A144280(n,m).

			a(4,3) = 9 = |S2(-3,2,1)|^2. The relevant partition of 4 is (2^2).

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

Cf. A036040, A143173, A134278, A000041, A008545.

Cf. A144274 (M32hat(-2) array), A144284 (M32hat(-4) array).

a(n,k) = Product_{j=1..n} |S2(-3,j,1)|^e(n,k,j), with |S2(-3,n,1)|= A008545(n-1) = (4*n-5)(!^4) (4-factorials) for n>=2 and 1 if 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.

Formally a(n,k)= 'M32(-3)/M3' = 'A143173/A036040' (elementwise division of arrays).

A144284 Partition number array, called M32hat(-4)= 'M32(-4)/M3'= 'A144267/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 4, 1, 36, 4, 1, 504, 36, 16, 4, 1, 9576, 504, 144, 36, 16, 4, 1, 229824, 9576, 2016, 1296, 504, 144, 64, 36, 16, 4, 1, 6664896, 229824, 38304, 18144, 9576, 2016, 1296, 576, 504, 144, 64, 36, 16, 4, 1, 226606464, 6664896, 919296, 344736, 254016, 229824, 38304, 18144

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) =: M32hat(-4;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,...].
If M32hat(-4;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-4):= A144285(n,m).

			a(4,3)= 16 = |S2(-4,2,1)|^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.

A144279 (M32hat(-3) array). A144341 (M32hat(-5) array)

a(n,k)= product(|S2(-4,j,1)|^e(n,k,j),j=1..n) with |S2(-4,n,1)|= A008546(n-1) = (5*n-6)(!^5) (5-factorials) for n>=2 and 1 if 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.

Formally a(n,k)= 'M32(-4)/M3' = 'A144267/A036040' (elementwise division of arrays).

A144269 Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 15, 3, 1, 1, 1, 105, 15, 3, 3, 1, 1, 1, 945, 105, 15, 9, 15, 3, 1, 3, 1, 1, 1, 10395, 945, 105, 45, 105, 15, 9, 3, 15, 3, 1, 3, 1, 1, 1, 135135, 10395, 945, 315, 225, 945, 105, 45, 15, 9, 105, 15, 9, 3, 1, 15, 3, 1, 3, 1, 1, 1, 2027025, 135135, 10395, 2835

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) =: M32hat(-1;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,...].
If M32hat(-1;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-1):= A144270(n,m).

			a(4,3)= 1 = |S2(-1,2,1)|^2. The relevant partition of 4 is (2^2).
[1]; [1,1]; [3,1,1]; [15,3,1,1,1]; [105,15,3,3,1,1,1]; ... [From _Wolfdieter Lang_, Oct 23 2008]

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

Cf. A144271 (M32hat(-2) array).

a(n,k)= product(|S2(-1,j,1)|^e(n,k,j),j=1..n) with |S2(-1,n,1)|= A001147(n-1) = (2*n-3)(!^2) (2-factorials) for n>=2 and 1 if 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.

Formally a(n,k)= 'M32(-1)/M3' = 'A143171/A036040' (elementwise division of arrays).

Corrected all entries. Wolfdieter Lang, Oct 23 2008

A144274 Partition number array, called M32hat(-2)= 'M32(-2)/M3'= 'A143172/A036040', related to A004747(n,m)= |S2(-2;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 2, 1, 10, 2, 1, 80, 10, 4, 2, 1, 880, 80, 20, 10, 4, 2, 1, 12320, 880, 160, 100, 80, 20, 8, 10, 4, 2, 1, 209440, 12320, 1760, 800, 880, 160, 100, 40, 80, 20, 8, 10, 4, 2, 1, 4188800, 209440, 24640, 8800, 6400, 12320, 1760, 800, 320, 200, 880, 160, 100, 40, 16, 80, 20

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) =: M32hat(-2;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,...].
If M32hat(-2;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-2):= A144275(n,m).

			a(4,3) = 4 = |S2(-2,2,1)|^2. The relevant partition of 4 is (2^2).

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

Cf. A144269 (M32hat(-1) array). A144279 (M32hat(-3) array).

a(n,k) = Product_{j=1..n} |S2(-2,j,1)|^e(n,k,j) with |S2(-2,n,1)|= A008544(n-1) = (3*n-4)(!^3) (3-factorials) for n>=2 and 1 if 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.

Formally a(n,k)= 'M32(-2)/M3' = 'A143172/A036040' (elementwise division of arrays).

A144341 Partition number array, called M32hat(-5)= 'M32(-5)/M3'= 'A144268/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

Original entry on oeis.org

1, 5, 1, 55, 5, 1, 935, 55, 25, 5, 1, 21505, 935, 275, 55, 25, 5, 1, 623645, 21505, 4675, 3025, 935, 275, 125, 55, 25, 5, 1, 21827575, 623645, 107525, 51425, 21505, 4675, 3025, 1375, 935, 275, 125, 55, 25, 5, 1, 894930575, 21827575, 3118225, 1182775, 874225, 623645

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) =: M32hat(-5;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,...].
If M32hat(-5;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-5):= A144342(n,m).

			a(4,3)= 25 = |S2(-5,2,1)|^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.

A144284 (M32hat(-4) array).

a(n,k)= product(|S2(-5,j,1)|^e(n,k,j),j=1..n) with |S2(-5,n,1)|= A008543(n-1) = (6*n-7)(!^6) (6-factorials) for n>=2 and 1 if 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.

Formally a(n,k)= 'M32(-5)/M3' = 'A144268/A036040' (elementwise division of arrays).

A264753 Irregular triangle read by rows: T(n,k) = A127671(n,k)/A036040(n,k), n >= 1 and 1 <= k <= A000041(n).

Original entry on oeis.org

1, 1, -1, 1, -1, 2, 1, -1, -1, 2, -6, 1, -1, -1, 2, 2, -6, 24, 1, -1, -1, -1, 2, 2, 2, -6, -6, 24, -120, 1, -1, -1, -1, 2, 2, 2, 2, -6, -6, -6, 24, 24, -120, 720, 1, -1, -1, -1, -1, 2, 2, 2, 2, 2, -6, -6, -6, -6, -6, 24, 24, 24, -120, -120, 720, -5040

Offset: 1

Johannes W. Meijer, Jul 12 2016

This sequence connects the multinomial coefficients A036040 (M_3) with A127671 (M_5).
The numbers of terms in n-th row is the number of partitions A000041(n). The number of terms T(n, k) with equal values in the n-th row follow the rhythm of A008284(n).
Some row sums are [1, 0, 2, -5, 21, -104, 636, -4511, 36455, -330954, 3334390, -36914039].

			The first few rows of the T(n,k) triangle:
n = 1: 1
n = 2: 1, -1
n = 3: 1, -1, 2
n = 4: 1, -1, -1, 2, -6
n = 5: 1, -1, -1, 2, 2, -6, 24
n = 6: 1, -1, -1, -1, 2, 2, 2, -6, -6, 24, -120
n = 7: 1, -1, -1, -1, 2, 2, 2, 2, -6, -6, -6, 24, 24, -120, 720

Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.

Cf. A036040 (M_3), A127671 (M_5), A000041, A008284, A081362.

Cf. A048996 (M_0), A036038 (M_1), A036039 (M_2), A117506 (M_4).

nmax:=8: with(combinat): A008284 := proc(n, k) option remember; if k < 0 or n < 0 then 0 elif k = 0 then if n = 0 then 1 else 0 fi else A008284(n-1, k-1) + A008284(n-k, k) fi end: for n from 1 to nmax do p:=0: k:=1: while k < numbpart(n)+1 do p := p+1: k1 := A008284(n, p): while k1 > 0 do A264753(n, k) := (-1)^(p+1)*(p-1)!: k := k+1: k1 := k1-1: od: od: od: seq(seq(A264753(n, k), k = 1..numbpart(n)), n = 1..nmax);

nMax = 8; A008284[n_, k_] := A008284[n, k] = If[k<0 || n<0, 0, If[k == 0, If[n == 0, 1, 0], A008284[n-1, k-1] + A008284[n-k, k]]]; For[n = 1, n <= nMax, n++, p = 0; k = 1; While[k < PartitionsP[n]+1, p = p+1; k1 = A008284[n, p]; While[k1>0, A264753[n, k] = (-1)^(p+1)*(p-1)!; k = k+1; k1 = k1-1]]]; Table[Table[A264753[n, k], {k, 1, PartitionsP[n]}], {n, 1, nMax}] // Flatten (* Jean-François Alcover, Oct 01 2016, translated from Maple *)

T(n, k) = A127671(n, k)/A036040(n, k), n >= 1 and 1 <= k <= A000041(n).

A122454 A triangle with shape A000041 defined by sequence A098546 times sequence A036040.

Original entry on oeis.org

1, 2, 1, 3, 9, 1, 4, 24, 18, 24, 1, 5, 50, 100, 100, 150, 50, 1, 6, 90, 225, 150, 300, 1200, 300, 300, 675, 90, 1, 7, 147, 441, 735, 735, 3675, 2450, 3675, 1225, 7350, 3675, 735, 2205, 147, 1, 8, 224, 784, 1568, 980, 1568, 9408, 15680, 11760, 15680, 3920, 29400

Offset: 1

Alford Arnold, Sep 18 2006

Shape sequence for A122454 is A000041 which counts numeric partitions.

			A098546(n) begins 1 2 1 3 3 1 4 6 6 4 1 ...
A036040(n) begins 1 1 1 1 3 1 1 4 3 6 1 ...
So the triangle begins:
1;
2,   1;
3,   9,   1;
4,  24,  18,  24,   1;
5,  50, 100, 100, 150,   50,    1;
6,  90, 225, 150, 300, 1200,  300,  300,  675,   90,    1;
7, 147, 441, 735, 735, 3675, 2450, 3675, 1225, 7350, 3675, 735, 2205, 147, 1;

Cf. A122455.

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: A098546 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then m := nops(op(k,prts)) ; binomial(n,m) ; else 0 ; fi ; end: M3 := proc(L) local n,k,an,resul; n := add(i,i=L) ; resul := factorial(n) ; for k from 1 to n do an := add(1-min(abs(j-k),1),j=L) ; resul := resul/ (factorial(k))^an /factorial(an) ; od ; end: A036040 := proc(n,k) local prts,m ; prts := combinat[partition](n) ; prts := sort(prts, sortAbrSteg) ; if k <= nops(prts) then M3(op(k,prts)) ; else 0 ; fi ; end: A122454 := proc(n,k) A098546(n,k)*A036040(n,k) ; end: for n from 1 to 10 do for k from 1 to combinat[numbpart](n) do a:=A122454(n,k) ; printf("%d, ",a) ; od; od ; # R. J. Mathar, Jul 17 2007

A122454(n) = A098546(n) times A036040(n).

More terms from R. J. Mathar, Jul 17 2007

cp's OEIS Frontend

A257490 Irregular triangle read by rows in which the n-th row lists multinomials (A036040) for partitions of 2n which have only even parts in Abramowitz-Stegun ordering.

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

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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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A144279 Partition number array, called M32hat(-3)= 'M32(-3)/M3'= 'A143173/A036040', related to A000369(n,m)= |S2(-3;n,m)| (generalized Stirling triangle).

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A144284 Partition number array, called M32hat(-4)= 'M32(-4)/M3'= 'A144267/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

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A144269 Partition number array, called M32hat(-1)= 'M32(-1)/M3'= 'A143171/A036040', related to A001497(n-1,m-1)= |S2(-1;n,m)| (generalized Stirling triangle).

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A144274 Partition number array, called M32hat(-2)= 'M32(-2)/M3'= 'A143172/A036040', related to A004747(n,m)= |S2(-2;n,m)| (generalized Stirling triangle).

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A144341 Partition number array, called M32hat(-5)= 'M32(-5)/M3'= 'A144268/A036040', related to A011801(n,m)= |S2(-4;n,m)| (generalized Stirling triangle).

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