A134134 -id:A134134 - cp's OEIS frontend

A144895 Second column of triangle A134134 (S2'(2) = S1hat(2)).

1, 2, 10, 36, 204, 1104, 7776, 57600, 505440, 4803840, 51442560, 597784320, 7609593600, 104364288000, 1541746483200, 24329797632000, 409042632499200, 7290954768384000, 137384159367168000, 2727604332085248000, 56913717580296192000, 1244955414746824704000

Offset: 0

Wolfdieter Lang, Oct 09 2008

Sum of the products of the factorials of the partition parts of n+2 into two parts. - Wesley Ivan Hurt, Mar 18 2016

			a(2)=10; The partitions of (2)+2 = 4 into two parts are: (3,1) and (2,2). The sum of the products of the factorials of the partition parts is: 3!*1! + 2!*2! = 6 + 4 = 10. - _Wesley Ivan Hurt_, Mar 18 2016

Cf. A000142 (factorials, first column). A144896 (third column).

Cf. A134134.

A144895:=n->add(i!*(n-i+2)!, i=1..floor(n/2)+1): seq(A144895(n), n=0..30); # Wesley Ivan Hurt, Mar 18 2016

Table[Sum[i!*(n - i + 2)!, {i, Floor[n/2] + 1}], {n, 0, 20}] (* Wesley Ivan Hurt, Mar 18 2016 *)

a(n) = A134134(n+2,2), n>=0.

a(n) = Sum_{i=1..floor(n/2)+1} i! * (n-i+2)!. - Wesley Ivan Hurt, Mar 18 2016

A134135 Alternating row sums of triangle A134134.

Original entry on oeis.org

1, 1, 5, 15, 93, 551, 4129, 33607, 312929, 3179343, 35602881, 432201743, 5678740945, 80142780751, 1210609725905, 19481112885231, 332836223507793, 6016678424942063, 114746996449871761, 2302527084416470255

Offset: 1

Wolfdieter Lang Oct 12 2007

Difference of numbers sum(product(j!^e(n,m,k,j),j=1..n),k=1..p(n,m)) related to odd and even part m partitions of n. Here e(n,m,k,j) is the exponent of j in the k-th m part partition of n and p(n,m)=A008284(n,m) is the number of partitions of n with m parts.

Cf. A077365 (row sums of A134134).

a(n)=sum(A134134(n,m)*(-1)^(m-1),m=1..n),n>=1.

A144896 Third column of triangle A134134 (S2'(2)= S1hat(2)).

Original entry on oeis.org

1, 2, 10, 44, 228, 1272, 8760, 63936, 547776, 5145984, 54233280, 624291840, 7879472640, 107423677440, 1579212910080, 24832164556800, 416273901926400, 7403098797158400, 139238590721126400, 2760253302701260800, 57522218527420416000, 1256931901812400128000

Offset: 0

Wolfdieter Lang, Oct 09 2008

Cf. A134134, A000142 (factorials, first column). A144895 (second column).

a(n) = A134134(n+3,3), n>=0.

A077365 Sum of products of factorials of parts in all partitions of n.

Original entry on oeis.org

1, 1, 3, 9, 37, 169, 981, 6429, 49669, 430861, 4208925, 45345165, 536229373, 6884917597, 95473049469, 1420609412637, 22580588347741, 381713065286173, 6837950790434781, 129378941557961565, 2578133190722896861, 53965646957320869469, 1183822028149936497501

Offset: 0

Vladeta Jovovic, Nov 30 2002

nonn

Row sums of arrays A069123 and A134133. Row sums of triangle A134134.

			The partitions of 4 are 4, 3+1, 2+2, 2+1+1, 1+1+1+1, the corresponding products of factorials of parts are 24,6,4,2,1 and their sum is a(4) = 37.
1 + x + 3 x^2 + 9 x^3 + 37 x^4 + 169 x^5 + 981 x^6 + 6429 x^7 + 49669 x^8 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 0..70 from Vincenzo Librandi)
J.-P. Bultel, A, Chouria, J.-G. Luque and O. Mallet, Word symmetric functions and the Redfield-Polya theorem, 2013.

Cf. A006906, A074141, A256125, A265950.
Cf. A069123, A134133, A134134.
Cf. A051296 (with compositions instead of partitions).

b:= proc(n, i, j) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, j)+
      `if`(i>n, 0, j^i*b(n-i, i, j+1))))
    end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 03 2013
# second Maple program:
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!)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 11 2016

Table[Plus @@ Map[Times @@ (#!) &, IntegerPartitions[n]], {n, 0, 20}] (* Olivier Gérard, Oct 22 2011 *)
a[ n_] := If[ n < 0, 0, Plus @@ Times @@@ (IntegerPartitions[ n] !)] (* Michael Somos, Feb 09 2012 *)
nmax=20; CoefficientList[Series[Product[1/(1-k!*x^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 14 2015 *)
b[n_, i_, j_] := b[n, i, j] = If[n==0, 1, If[i<1, 0, b[n, i-1, j] + If[i>n, 0, j^i*b[n-i, i, j+1]]]]; a[n_] := b[n, n, 1]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

N=66; q='q+O('q^N);
gf= 1/prod(n=1,N, (1-n!*q^n) );
Vec(gf)
/* Joerg Arndt, Oct 06 2012 */

G.f.: 1/Product_{m>0} (1-m!*x^m).
Recurrence: a(n) = 1/n*Sum_{k=1..n} b(k)*a(n-k), where b(k) = Sum_{d divides k} d*d!^(k/d).
a(n) ~ n! * (1 + 1/n + 3/n^2 + 12/n^3 + 67/n^4 + 457/n^5 + 3734/n^6 + 35741/n^7 + 392875/n^8 + 4886114/n^9 + 67924417/n^10), for coefficients see A256125. - Vaclav Kotesovec, Mar 14 2015
G.f.: exp(Sum_{k>=1} Sum_{j>=1} (j!)^k*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

Unnecessarily complicated mma code deleted by N. J. A. Sloane, Sep 21 2009

A134133 A certain partition array in Abramowitz-Stegun order (A-St order).

Original entry on oeis.org

1, 2, 1, 6, 2, 1, 24, 6, 4, 2, 1, 120, 24, 12, 6, 4, 2, 1, 720, 120, 48, 36, 24, 12, 8, 6, 4, 2, 1, 5040, 720, 240, 144, 120, 48, 36, 24, 24, 12, 8, 6, 4, 2, 1, 40320, 5040, 1440, 720, 576, 720, 240, 144, 96, 72, 120, 48, 36, 24, 16, 24, 12, 8, 6, 4, 2, 1, 362880, 40320, 10080

Offset: 1

Wolfdieter Lang, Oct 12 2007

The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].

Partition number array M_3(2)= A130561 divided by partition number array M_3 = M_3(1) = A036040.

			[1], [2,1], [6,2,1], [24,6,4,2,1], [120,24,12,6,4,2,1], ...

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].
Wolfdieter Lang, First 10 rows and more.

With another ordering of the partitions this becomes A069123.

Cf. A134134 (triangle obtained by summing same m numbers).

a(n,k) = A130561(n,k)/A036040(n,k) (division of partition arrays M_3(2) by M_3).

a(n,k) = product(j!^e(n,k,j),j=1..n) with 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.

A134146 Triangle of numbers obtained from the partition array A134145.

Original entry on oeis.org

1, 3, 1, 15, 3, 1, 105, 24, 3, 1, 945, 150, 24, 3, 1, 10395, 1485, 177, 24, 3, 1, 135135, 14805, 1620, 177, 24, 3, 1, 2027025, 191520, 16425, 1701, 177, 24, 3, 1, 34459425, 2687580, 208125, 16830, 1701, 177, 24, 3, 1, 654729075, 44552025, 2880360, 212985

Offset: 1

Wolfdieter Lang, Nov 13 2007

This triangle is named S2(3)'.

In the same manner the unsigned Lah triangle A008297 is obtained from the partition array A130561.

			[1]; [3,1]; [15,3,1]; [105,24,3,1]; [945,150,24,3,1];...

W. Lang, First 10 rows and more.

Cf. A134147 (row sums).
Cf. A134148 (allternating row sums).
Cf. A134134 (k=2 member of this triangle family).

a(n,m)=sum(product(S2(3;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. S2(3;j,1)= A001147(j) = A035342(j,1) = (2*j-1)!!.

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)!.

cp's OEIS Frontend

A144895 Second column of triangle A134134 (S2'(2) = S1hat(2)).

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A134135 Alternating row sums of triangle A134134.

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A144896 Third column of triangle A134134 (S2'(2)= S1hat(2)).

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A077365 Sum of products of factorials of parts in all partitions of n.

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A134133 A certain partition array in Abramowitz-Stegun order (A-St order).

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A134146 Triangle of numbers obtained from the partition array A134145.

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

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