A283154 -id:A283154 - cp's OEIS frontend

A090209 Generalized Bell numbers (from (5,5)-Stirling2 array A090216).

1, 1, 1546, 12962661, 363303011071, 25571928251231076, 3789505947767235111051, 1049433111253356296672432821, 498382374325731085522315594481036, 380385281554629647028734545622539438171, 443499171330317702437047276255605780991365151

Offset: 0

Wolfdieter Lang, Dec 01 2003

Contribution from Peter Luschny, Mar 27 2011: (Start) Let B_{m}(x) = sum_{j>=0}(exp(j!/(j-m)!*x-1)/j!) then a(n) = n! [x^n] taylor(B_{5}(x)), where [x^n] denotes the coefficient of x^k in the Taylor series for B_{5}(x).

a(n) is row 5 of the square array representation of A090210. (End)

M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

G. C. Greubel, Table of n, a(n) for n = 0..115
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem., arXiv:quant-ph/0402027, Phys. Lett. A 309 (3-4) (2003) 198-205
K. A. Penson, P. Blasiak, A. Horzela, A. I. Solomon and G. H. E. Duchamp, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. 50, 083512 (2009).

Cf. (Generalized) Bell numbers from (m,m)-Stirling2 array: A000110 (m=1), A020556 (m=2), A069223 (m=3), A071379 (m=4). Triangle A090210.

A071379 := proc(n) local r,s,i;
if n=0 then 1 else r := [seq(6,i=1..n-1)]; s := [seq(1,i=1..n-1)];
exp(-x)*5!^(n-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(A071379(n),n=0..8); # Peter Luschny, Mar 30 2011

fallfac[n_, k_] := Pochhammer[n-k+1, k]; a[n_, k_] := (((-1)^k)/k!)*Sum[((-1)^p)*Binomial[k, p]*fallfac[p, 5]^n, {p, 5, k}]; a[0] = 1; a[n_] := Sum[a[n, k], {k, 5, 5*n}];  Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Mar 05 2014 *)

a(n) = Sum_{k=5..5*n} A090216(n, k), n>=1. a(0) := 1.
a(n) = Sum_{k >=5} (fallfac(k, 5)^n)/k!/exp(1), n>=1, a(0) := 1. From eq.(26) with r=5 of the Schork reference.
E.g.f. with a(0) := 1: (sum((exp(fallfac(k, 5)*x))/k!, k=5..infinity)+ A000522(4)/4!)/exp(1). From the top of p. 4656 with r=5 of the Schork reference.

If it is proved that A283154 and A090209 are the same, then the entries should be merged and A283154 recycled. - N. J. A. Sloane, Mar 06 2017

A283153 Number of set partitions of unique elements from an n X 4 matrix where elements from the same row may not be in the same partition.

Original entry on oeis.org

1, 209, 163121, 326922081, 1346634725665, 9939316337679281, 119802044788535500753, 2205421644124274191535553, 58945667435045762187763602753, 2198513228897522394476415669503377, 110833342180980170285766876408530089329, 7356710448423295420590529054176924329802529, 628972339934967292421997567343442748145219556449

Offset: 1

Marko Riedel, Mar 01 2017

nonn

Apparently a duplicate of A071379? - R. J. Mathar, Mar 06 2017

Cf. A283154, A283155, A071379.

Table[(4!^n) * Sum[Binomial[p,4]^n/p! * Sum[(-1)^k/k!,{k,0,4n-p}],{p,1,4n}],{n,1,50}] (* Indranil Ghosh, Mar 04 2017 *)

a(n) = (4!^n) * sum(p=1, 4*n, binomial(p,4)^n/p! * sum(k=0, 4*n-p, (-1)^k/k!)); \\ Indranil Ghosh, Mar 04 2017

a(n) = m!^n * Sum_{p=1..n*m} (binomial(p,m)^n/p!) * Sum_{k=0..n*m-p} (-1)^k/k! with m=4.

If it is proved that A283153 and A071379 are the same, then the entries should be merged and A283153 recycled. - N. J. A. Sloane, Mar 06 2017

A283155 Number of set partitions of unique elements from an n X 6 matrix where elements from the same row may not be in the same partition.

Original entry on oeis.org

1, 13327, 1395857215, 637056434385865, 893591647147188285577, 3104750712141723393459934903, 23094793819000630529788087185212647, 331114050237261411471736187067402011971825, 8452444659410086110360476363825233533247222327537, 361084373753302872550305348321621374196786909194880037375

Offset: 1

Marko Riedel, Mar 01 2017

nonn

Indranil Ghosh, Table of n, a(n) for n = 1..50
M. Riedel, Set partitions of unique elements from an n-by-m matrix where elements from the same row may not be in the same partition

Cf. A283153, A283154.

Table[(6!^n)*Sum[Binomial[p,6]^n/p! * Sum[(-1)^k/k!,{k,0,6n-p}],{p,1,6n}],{n,1,10}] (* Indranil Ghosh, Mar 04 2017 *)

a(n) = (6!^n)*sum(p=1,6*n,binomial(p,6)^n/p! * sum(k=0,6*n-p,(-1)^k/k!)); \\ Indranil Ghosh, Mar 04 2017

a(n) = m!^n Sum_{p=1..n*m} (Choose(p,m)^n/p!) Sum_{k=0..n*m-p} (-1)^k/k! with m=6.

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A090209 Generalized Bell numbers (from (5,5)-Stirling2 array A090216).

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A283153 Number of set partitions of unique elements from an n X 4 matrix where elements from the same row may not be in the same partition.

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A283155 Number of set partitions of unique elements from an n X 6 matrix where elements from the same row may not be in the same partition.

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