A090210 Triangle of certain generalized Bell numbers.

1, 1, 1, 2, 1, 1, 5, 7, 1, 1, 15, 87, 34, 1, 1, 52, 1657, 2971, 209, 1, 1, 203, 43833, 513559, 163121, 1546, 1, 1, 877, 1515903, 149670844, 326922081, 12962661, 13327, 1, 1, 4140, 65766991, 66653198353, 1346634725665, 363303011071, 1395857215, 130922, 1, 1

Offset: 1

Author

Wolfdieter Lang, Dec 01 2003

Keywords

• nonn
• easy
• tabl

Comments

Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) and

S(n,k) = k! [x^k] taylor(B_{n}(x)), where [x^k] denotes the

coefficient of x^k in the Taylor series for B_{n}(x).

Then S(n,k) (n>0, k>=0) is the square array representation of the triangle.

To illustrate the cross-references of T(n,k) when written as a square array.

n\k A000012,A000012,A002720,A069948,A182925,A182924,...

0: A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...

1: A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...

2: A020556: 1, 1, 7, 87, 1657, 43833, 1515903, ...

3: A069223: 1, 1, 34, 2971, 513559, 149670844, ...

4: A071379: 1, 1, 209, 163121, 326922081, ...

5: A090209: 1, 1, 1546, 12962661, 363303011071,...

6: ... 1, 1, 13327, 1395857215, 637056434385865,...

Note that the sequence T(0,k) is not included in the data.

- Peter Luschny, Mar 27 2011

Examples

			Triangle begins:
1;
1, 1;
2, 1, 1;
5, 7, 1, 1;
15, 87, 34, 1, 1;
52, 1657, 2971, 209, 1, 1;
203, 43833, 513559, 163121, 1546, 1, 1;
		

References

• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
• M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

Links

• P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
• W. Lang, First 8 rows.
• 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).

Crossrefs

Cf. A000110, A020556, A069223, A071379, A090209, A002720, A069948, A182924, A182925, A182933.

T(n,n) gives A070227.

Programs

Maple

A090210_AsSquareArray := proc(n,k) local r,s,i;
if k=0 then 1 else r := [seq(n+1,i=1..k-1)]; s := [seq(1,i=1..k-1)];
exp(-x)*n!^(k-1)*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) fi end:
seq(lprint(seq(A090210_AsSquareArray(n,k),k=0..6)),n=0..6);
# Peter Luschny, Mar 30 2011

Mathematica

t[n_, k_] := t[n, k] = Sum[(n+j)!^(k-1)/(j!^k*E), {j, 0, Infinity}]; t[_, 0] = 1;
Flatten[ Table[ t[n-k+1, k], {n, 0, 8}, {k, n, 0, -1}]][[1 ;; 43]] (* Jean-François Alcover, Jun 17 2011 *)

Formula

a(n, m) = Bell(m;n-(m-1)), n>= m-1 >=0, with Bell(m;k) := Sum_{p=m..m*k} S2(m;k, p), where S2(m;k, p) := (((-1)^p)/p!) * Sum_{r=m..p} ((-1)^r)*binomial(p, r)*fallfac(r, m)^k; with fallfac(n, m) := A008279(n, m) (falling factorials) and m<=p<=k*m, k>=1, m=1, 2, ..., else 0. From eqs.(6) with r=s->m and eq.(19) with S_{r, r}(n, k)-> S2(r;n, k) of the Blasiak et al. reference. [Corrected by Sean A. Irvine, Jun 03 2024]

a(n, m) = (Sum_{k>=m} fallfac(k, m)^(n-(m-1)))/exp(1), n>=m-1>=0, else 0. From eq.(26) with r->m of the Schork reference which is rewritten eq.(11) of the original Blasiak et al. reference.

E.g.f. m-th column (no leading zeros): (Sum_{k>=m} exp(fallfac(k, m)*x)/k!) + A000522(m)/m!)/exp(1). Rewritten from the top of p. 4656 of the Schork reference.