%I A049020
%S A049020 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,
%T A049020 155,21,1,877,3263,4802,3556,1400,287,28,1,4140,17007,28337,24626,
%U A049020 11991,3290,490,36,1,21147,94828,175896,174805,101031,34671,6972,786
%N A049020 Triangle of numbers a(n,k), 0<=k<=n, related to Bell numbers.
%C A049020 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.
%C A049020 Exponential Riordan array [exp(exp(x)-1), exp(x)-1]. [From Paul Barry 
               (pbarry(AT)wit.ie), Jan 12 2009]
%D A049020 M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 
               (1999), 207-210.
%D A049020 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 (njas(AT)research.att.com), 
               Feb 07 2009]
%F A049020 a(n, k)=a(n-1, k-1)+(k+1)*a(n-1, k)+(k+1)*a(n-1, k+1), n >= 1.
%F A049020 a(n, k)=Sum_{i=0..n} stirling2(n, i)*binomial(i, k), k=0..n. E.g.f. for 
               k-th column is (1/k!) *(exp(x)-1)^k*exp(exp(x)-1) - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Jan 27 2001
%F A049020 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 (pbarry(AT)wit.ie), 
               Apr 29 2009]
%e A049020 Triangle begins:
%e A049020 1;
%e A049020 1,1;
%e A049020 2,3,1;
%e A049020 5,10,6,1;
%e A049020 15,37,31,10,1;
%e A049020 ...
%e A049020 Contribution from Paul Barry (pbarry(AT)wit.ie), Jan 12 2009: (Start)
%e A049020 Production array begins
%e A049020 1,1,
%e A049020 1,2,1,
%e A049020 0,2,3,1,
%e A049020 0,0,3,4,1,
%e A049020 0,0,0,4,5,1 (End)
%o A049020 (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))
%Y A049020 First column gives A000110, second column = A005493.
%Y A049020 Third column = A003128, row sums = A001861, A059340.
%Y A049020 Sequence in context: A090299 A060693 A089302 this_sequence A144634 A147315 
               A085853
%Y A049020 Adjacent sequences: A049017 A049018 A049019 this_sequence A049021 A049022 
               A049023
%K A049020 nonn,tabl,nice,easy
%O A049020 0,4
%A A049020 N. J. A. Sloane (njas(AT)research.att.com).
%E A049020 More terms from James A. Sellers (sellersj(AT)math.psu.edu)