A182925 -id:A182925 - cp's OEIS frontend

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

Wolfdieter Lang, Dec 01 2003

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

			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;

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.

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).

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

T(n,n) gives A070227.

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

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 *)

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.

A182924 Generalized vertical Bell numbers of order 4.

Original entry on oeis.org

1, 52, 43833, 149670844, 1346634725665, 25571928251231076, 893591647147188285577, 52327970757667659912764908, 4796836032234830356783078467969, 653510798275634770675047022800897940, 127014654376520087360456517007106313763801

Offset: 0

Peter Luschny, Mar 28 2011

nonn

The name "generalized 'vertical' Bell numbers" is used to distinguish them from the generalized (horizontal) Bell numbers with reference to the square array representation of the generalized Bell numbers as given in A090210. a(n) is column 5 in this representation. The order is the parameter M in Penson et al., p. 6, eq. 29.

Apparently a(n) = A157280(n+1) for 0 <= n <= 8. - Georg Fischer, Oct 24 2018 (and true considering the hypergeometric comment in A157280, R. J. Mathar, Apr 23 2024).

G. C. Greubel, Table of n, a(n) for n = 0..129
P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation, arXiv:1010.0354 [math.CO], 2010-2011.
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. A090210, A002720, A069948,A157280, A182925.

A182924 := proc(n) exp(-x)*GAMMA(n+1)^4*hypergeom([n+1,n+1,n+1,n+1],[1,1,1,1],x): simplify(subs(x=1, %)) end;
seq(A182924(i),i=0..10);

fallfac[n_, k_] := Pochhammer[n-k+1, k]; f[m_][n_, k_] := (-1)^k/k!* Sum[(-1)^p*Binomial[k, p]*fallfac[p, m]^n, {p, m, k}]; a[n_] := Sum[f[n][5, k], {k, n, 5*n}]; Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Sep 05 2012 *)

a(n) = exp(-1)*Gamma(n+1)^4*[4F4]([n+1,n+1,n+1,n+1], [1,1,1,1] | 1); here [4F4] is the generalized hypergeometric function of type 4F4.

Let B_{n}(x) = sum_{j>=0}(exp(j!/(j-n)!*x-1)/j!) then a(n) = 5! [x^5] taylor(B_{n}(x)), where [x^5] denotes the coefficient of x^5 in the Taylor series for B_{n}(x).

A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 5, 27, 13, 1, 1, 15, 409, 778, 73, 1, 1, 52, 9089, 104149, 37553, 501, 1, 1, 203, 272947, 25053583, 57184313, 2688546, 4051, 1, 1, 877, 10515147, 9566642254, 192052025697, 56410245661, 265141267, 37633, 1

Offset: 0

Peter Luschny, Mar 29 2011

These numbers are related to the generalized Bell numbers based on the falling factorial powers (A090210).
The square array starts for n>=0, k>=0:
n\k=0,1,..  A000012,A000262,A182934,...
A000012: 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
A000110: 1, 1, 2, 5, 15, 52, 203, 877, 4140, ...
A094577: 1, 3, 27, 409, 9089, 272947, 10515147, ...
A182932: 1, 13, 778, 104149, 25053583, 9566642254, ...
       : 1, 73, 37553, 57184313, 192052025697, ...
       : 1, 501, 2688546, 56410245661, ...
 ....  : 1, 4051, 265141267, 89501806774945, ...

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

A182933_AsSquareArray := proc(n,k) local r,s,i;
r := [seq(n+1,i=1..k)]; s := [seq(1,i=1..k-1),2];
exp(-x)*n!^k*hypergeom(r,s,x); round(evalf(subs(x=1,%),99)) end:
seq(lprint(seq(A182933_AsSquareArray(n,k),k=0..6)),n=0..6);

a[n_, k_] := Exp[-1]*n!^k*HypergeometricPFQ[ Table[n+1, {k}], Append[ Table[1, {k-1}], 2], 1.]; Table[ a[n-k, k] // Round , {n, 0, 8}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

Let r_k = [n+1,...,n+1] (k occurrences of n+1), s_k = [1,...,1,2] (k-1 occurrences of 1) and F_k the generalized hypergeometric function of type k_F_k, then a(n,k) = exp(-1)*n!^k*F_k(r_k, s_k | 1).

Let B_{n}(x) = sum_{j>=0}(exp((j+n-1)!/(j-1)!*x-1)/j!) then a(n,k) = k! [x^k] series(B_{n}(x)), where [x^k] denotes the coefficient of x^k in the Taylor series for B_{n}(x).

cp's OEIS Frontend

A090210 Triangle of certain generalized Bell numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A182924 Generalized vertical Bell numbers of order 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A182933 Generalized Bell numbers based on the rising factorial powers; square array read by antidiagonals.

Views

Author

Keywords

Comments

Crossrefs

Programs

Maple

Mathematica

Formula