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A090210
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Triangle of certain generalized Bell numbers.
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11
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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
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OFFSET
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1,4
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COMMENTS
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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.
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.
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REFERENCES
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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.
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LINKS
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FORMULA
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a(n, m)= Bell(m;n-(m-1)), n>= m-1 >=0, with Bell(m;k) := sum(S2(m;k, p), p=m..m*k), where S2(m;k, p) := (((-1)^p)/p!)*sum(((-1)^r)*binomial(p, r)*fallfac(p, r)^k, r=m..p); 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.
a(n, m)= (sum(fallfac(k, m)^(n-(m-1)), k=m..infinity))/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((exp(fallfac(k, m)*x))/k!, k=m..infinity) + A000522(m)/m!)/exp(1). Rewritten from the top of p. 4656 of the Schork reference.
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EXAMPLE
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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;
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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