0,4

Or, triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,2,3,3,4,4,5,5,6,6,...] DELTA [1,0,1,0,1,0,1,0,1,0,1,0,...] where DELTA is the operator defined in A084938 .

Reversal of A094638 .

Equals A132393*A007318, as infinite lower triangular matrices . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 13 2007

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)

The higher order exponential integrals E(x,m,n) are defined in A163931. The asymptotic expansion of the exponential integrals E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + n*(n+1)/x^2 - n*(n+1)*(n+2)/x^3 + .. ), see Abramowitz and Stegun. This formula follows from the general formula for the asymptotic expansion, see A163932. We rewrite E(x,m=1,n) ~ (exp(-x)/x)*(1 - n/x + (n^2+n)/x^2 - (2*n+3*n^2+n^3)/x^3 + (6*n+11*n^2+6*n^3+n^4)/x^3 - .. ) and observe that the T(n,m) are the polynomials coefficients in the denominators. Looking at the a(n,m) formula of A028421, A163932 and A163934, and shifting the offset given above to 1, we can write T(n-1,m-1) = a(n,m) = (-1)^(n+m)*stirling1(n,m), see the Maple program.

The asymptotic expansion leads for values of n from one to eleven to known sequences, see the cross-references. With these sequences one can form the triangles A008279 (right hand columns) and A094587 (left hand columns).

See A163936 for information about the o.g.f.s. of the right hand columns of this triangle.

(End)

T. D. Noe, Rows n=0..50 of triangle, flattened

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, Chapter 5, pp. 227-251. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009]

T(0,0)=1, T(k,n)=0 if k>n or if n<0, T(n,k)=T(n-1,k-1)+n*T(n-1,k). T(n,0)=n!=A000142(n). T(2*n,n)=A129505(n+1). Sum_{k, 0<=k<=n}T(n,k)=(n+1)!=A000142(n+1). Sum_{k, 0<=k<=n}T(n,k)^2=A047796(n+1). T(n,k)=|Stirling1(n+1,k+1)|, see A008275 . (x+1)(x+2)...(x+n)=Sum_{k, 0<=k<=n}T(n,k)*x^k. [Corrected by Arie Bos, Jul 11 2008]

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 13 2007

Triangle begins:

1;

1, 1;

2, 3, 1;

6, 11, 6, 1;

24, 50, 35, 10, 1;

120, 274, 225, 85, 15, 1;

720, 1764, 1624, 735, 175, 21, 1;

5040, 13068, 13132, 6769, 1960, 322, 28, 1;

40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1;

362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1 ;...

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)

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

(End)

Diagonals : A000012 A000217 A000914 A001303 A000915 A053567 A112002. Columns A000142 A000254 A000399 A000454 A000482 A001233 A001234.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 07 2009: (Start)

Row sums equal A000142.

The asymptotic expansions lead to A000142 (n=1), A000142(n=2; minus a(0)), A001710 (n=3), A001715 (n=4), A001720 (n=5), A001725 (n=6), A001730 (n=7), A049388 (n=8), A049389 (n=9), A049398 (n=10), A051431 (n=11), A008279 and A094587.

Cf. A163931 (E(x,m,n)), A028421 (m=2), A163932 (m=3), A163934 (m=4), A163936.

(End)

Sequence in context: A138771 A121748 A008275 this_sequence A107416 A105613 A135894

Adjacent sequences: A130531 A130532 A130533 this_sequence A130535 A130536 A130537

nonn,tabl

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 09 2007