0,4

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 (pbarry(AT)wit.ie), Jan 12 2009]

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 (njas(AT)research.att.com), Feb 07 2009]

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), 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

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]

Triangle begins:

1;

1,1;

2,3,1;

5,10,6,1;

15,37,31,10,1;

...

Contribution from Paul Barry (pbarry(AT)wit.ie), 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)

(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))

First column gives A000110, second column = A005493.

Third column = A003128, row sums = A001861, A059340.

Sequence in context: A090299 A060693 A089302 this_sequence A144634 A147315 A085853

Adjacent sequences: A049017 A049018 A049019 this_sequence A049021 A049022 A049023

nonn,tabl,nice,easy

N. J. A. Sloane (njas(AT)research.att.com).

More terms from James A. Sellers (sellersj(AT)math.psu.edu)