A216379
Triangle of generalized Stirling numbers S_{n,n}(5,k) read by rows (n>=0, n<=k<=5n) the sum of which is A182924.
Original entry on oeis.org
1, 1, 15, 25, 10, 1, 16, 1280, 9080, 16944, 12052, 3840, 580, 40, 1, 1296, 330480, 6148872, 28245672, 49658508, 41392620, 18428400, 4691412, 706833, 63375, 3285, 90, 1, 331776, 207028224, 8190885888, 74684104704, 253100173824, 405044582400, 351783415296, 181005401088, 58436640576, 12288192000, 1721191680, 162115584, 10228144, 423360, 10960, 160, 1
Offset: 0
{1},
{1,15,25,10,1},
{16,1280,9080,16944,12052,3840,580,40,1}
...
Second row (n=1) is 5th row of
A008277 (Stirling numbers S2).
Third row is 5th row of
A078739 (Generalized Stirling numbers S_{2,2}).
Fourth row is 5th row of
A078741 (Generalized Stirling numbers S_{3,3}).
Fifth row is 5th row of
A090214 (Generalized Stirling numbers S_{4,4}).
-
f[m_][n_, k_] := (-1)^k/k!*Sum[(-1)^p*Binomial[k, p]*FactorialPower[p, m]^n, {p, m, k}]; Table[f[n][5, k],{n,0,4}, {k, n, 5*n}]//Flatten
A090210
Triangle of certain generalized Bell numbers.
Original entry on oeis.org
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
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.
-
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 *)
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
-
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 *)
A182925
Generalized vertical Bell numbers of order 3.
Original entry on oeis.org
1, 15, 1657, 513559, 326922081, 363303011071, 637056434385865, 1644720885001919607, 5943555582476814384769, 28924444943026683877502191, 183866199607767992029159792281, 1489437787210535537087417039489815
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..168
- P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation, (2010).
- 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).
-
A182925 := proc(n) exp(-x)*GAMMA(n+1)^3*hypergeom([n+1,n+1,n+1],[1,1,1],x);
round(evalf(subs(x=1,%),64)) end; seq(A182925(i),i=0..11);
-
u = 1.`64; a[n_] := n!^3*HypergeometricPFQ[{n+u, n+u, n+u}, {u, u, u}, u]/E // Round; Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Nov 22 2012, after Maple *)
Showing 1-4 of 4 results.
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