1,4

The asymptotic expansions of the higher order exponential integral E(x,m=1,n) lead to triangle A130524, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A130534 have a nice structure Gf(p) = W1(z,p)/(1-z)^(2*p-1) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc.. The coefficients of the W1(z,p) polynomials lead to the triangle given above, n =>1 and 1<= m <= n. Our triangle is the same as A112007 with an extra right hand column, see also the second Eulerian triangle A008517. The row sums of our triangle lead to A001147.

We observe that the row sums of the triangles A163936 (m=1), A163937 (m=2), A163938 (m=3) and A163939 (m=4) for z=1 lead to A001147, A001147 (minus a(0)), A001879 and A000457 which are the first four left hand columns of the triangle of the Bessel coefficients A001497 or if one wishes the right hand columns of A001498. We checked this phenomenon for a few more values of m and found that this pattern persists: m = 5 leads to A001880, m=6 to A001881, m=7 to A038121 and m=8 to A130563 which are the next left (right) hand columns of A001497 (A001498). An interesting phenomenom.

a(n,m) = sum((-1)^(n+k+1)*binomial(2*n-1,k)*stirling1(m+n-k-1,m-k),k=0..m-1)

The first few W1(z,p) polynomials are:

W1(z,p=1) = 1/(1-z)

W1(z,p=2) = (1+0*z)/(1-z)^3

W1(z,p=3) = (2+z+0*z^2)/(1-z)^5

W1(z,p=4) = (6+8*z+z^2+0*z^3)/(1-z)^7

with(combinat, stirling1): nmax:=10; for n from 1 to nmax do for m from 1 to n do a(n, m):= sum((-1)^(n+k+1)*binomial(2*n-1, k)*stirling1(m+n-k-1, m-k), k=0..m-1) od: od: T:=1: for n from 1 to nmax do for m from 1 to n do a(T):=a(n, m): T:=T+1: od: od: seq(a(n), n=1..T-1);

Row sums equal A001147.

A000142, A002538, A002539, A112008, A112485 are the first few left hand columns.

A000007, A000012, A005803(n+2), A004301, A006260 are the first few right hand columns.

Cf. A163931 (E(x,m,n)), A048994 (Stirling1) and A008517 (Euler).

Cf. A112007, A163937 (E(x,m=2,n)), A163938 (E(x,m=3,n)) and A163939 (E(x,m=4,n)).

Cf. A001497 (Bessel), A001498 (Bessel), A001147 (m=1), A001147 (m=2), A001879 (m=3) and A000457 (m=4), A001880 (m=5), A001881 (m=6) and A038121 (m=7).

Sequence in context: A111184 A111596 A129062 this_sequence A117651 A109971 A021896

Adjacent sequences: A163933 A163934 A163935 this_sequence A163937 A163938 A163939

easy,nonn,tabl

Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 13 2009