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A049020
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Triangle of numbers a(n,k), 0<=k<=n, related to Bell numbers.
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13
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1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 15, 37, 31, 10, 1, 52, 151, 160, 75, 15, 1, 203, 674, 856, 520, 155, 21, 1, 877, 3263, 4802, 3556, 1400, 287, 28, 1, 4140, 17007, 28337, 24626, 11991, 3290, 490, 36, 1, 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Triangle a(n,k) read by rows; given by [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1,...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deleham's operator defined in A084938.
Exponential Riordan array [exp(exp(x)-1), exp(x)-1]. [From Paul Barry, Jan 12 2009]
Equal to A048993*A007318. - From DELEHAM Philippe, Oct 31 2011
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REFERENCES
| M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16. [From N. J. A. Sloane, Feb 07 2009]
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FORMULA
| a(n, k) = a(n-1, k-1)+(k+1)*a(n-1, k)+(k+1)*a(n-1, k+1), n >= 1.
a(n, k) = sum(i=0..n, stirling2(n, i)*binomial(i, k) ). E.g.f. for the k-th column is (1/k!) *(exp(x)-1)^k*exp(exp(x)-1) - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 27 2001
G.f.: 1/(1-x-xy-x^2(1+y)/(1-2x-xy-2x^2(1+y)/(1-3x-xy-3x^2(1+y)/(1-4x-xy-4x^2(1+y)/(1-... (continued fraction). [From Paul Barry, Apr 29 2009]
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EXAMPLE
| Triangle begins:
1;
1,1;
2,3,1;
5,10,6,1;
15,37,31,10,1;
...
Contribution from Paul Barry, Jan 12 2009: (Start)
Production array begins
1,1,
1,2,1,
0,2,3,1,
0,0,3,4,1,
0,0,0,4,5,1 (End)
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MATHEMATICA
| a[n_, k_] = Sum[StirlingS2[n, i]*Binomial[i, k], {i, 0, n}]; Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]]
(* From Jean-François Alcover, Aug 29 2011, after V. Jovovic *)
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PROG
| (PARI) T(n, k)=if(k<0|k>n, 0, n!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)), n), k))
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CROSSREFS
| First column gives A000110, second column = A005493.
Third column = A003128, row sums = A001861, A059340.
Sequence in context: A060693 A172381 A089302 * A144634 A178125 A147315
Adjacent sequences: A049017 A049018 A049019 * A049021 A049022 A049023
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KEYWORD
| nonn,tabl,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu)
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