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A163937 Triangle related to the o.g.f.s. of the right-hand columns of A028421 (E(x,m=2,n)). 5
1, 1, 2, 2, 10, 3, 6, 52, 43, 4, 24, 308, 472, 136, 5, 120, 2088, 4980, 2832, 369, 6, 720, 16056, 53988, 49808, 13638, 918, 7, 5040, 138528, 616212, 826160, 381370, 57540, 2167, 8, 40320, 1327392, 7472952, 13570336, 9351260, 2469300, 222908, 4948, 9 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The asymptotic expansions of the higher-order exponential integral E(x,m=2,n) lead to triangle A028421, see A163931 for information on E(x,m,n). The o.g.f.s. of the right-hand columns of triangle A028421 have a nice structure: gf(p) = W2(z,p)/(1-z)^(2*p) with p = 1 for the first right-hand column, p = 2 for the second right-hand column, etc. The coefficients of the W2(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. The row sums of this triangle lead to A001147 (minus a(0)), see A163936 for more information.

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

a(n,m) = Sum_{k=0..(m-1)} (-1)^(n+k+1)*((m-k)/1!)*binomial(2*n,k)*Stirling1(m+n-k-1,m-k), 1 <= m <= n.

EXAMPLE

The first few W2(z,p) polynomials are

W2(z,p=1) = 1/(1-z)^2;

W2(z,p=2) = (1 +  2*z)/(1-z)^4;

W2(z,p=3) = (2 + 10*z +  3*z^2)/(1-z)^6;

W2(z,p=4) = (6 + 52*z + 43*z^2 + 4*z^3)/(1-z)^8.

MAPLE

with(combinat): a := proc(n, m): add((-1)^(n+k+1)*((m-k)/1!)*binomial(2*n, k)*stirling1(m+n-k-1, m-k), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..9);  # Johannes W. Meijer, revised Nov 27 2012

MATHEMATICA

Table[Sum[(-1)^(n + k + 1)*((m - k)/1!)*Binomial[2*n, k]*StirlingS1[m + n - k - 1, m - k], {k, 0, m - 1}], {n, 1, 10}, {m, 1, n}] // Flatten (* G. C. Greubel, Aug 13 2017 *)

PROG

(PARI) for(n=1, 10, for(m=1, n, print1(sum(k=0, m-1, (-1)^(n+k+1)*((m-k)/1!)*binomial(2*n, k) *stirling1(m+n-k-1, m-k)), ", "))) \\ G. C. Greubel, Aug 13 2017

CROSSREFS

Row sums equal A001147 (n>=1).

A000142, 2*A001705, are the first two left hand columns.

A000027 is the first right hand column.

Cf. A163931 (E(x,m,n)) and A028421.

Cf. A163936 (E(x,m=1,n)), A163938 (E(x,m=3,n)) and A163939 (E(x,m=4,n)).

Sequence in context: A321415 A319692 A308693 * A083457 A163808 A223126

Adjacent sequences:  A163934 A163935 A163936 * A163938 A163939 A163940

KEYWORD

easy,nonn,tabl

AUTHOR

Johannes W. Meijer, Aug 13 2009

STATUS

approved

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Last modified March 5 15:13 EST 2021. Contains 341823 sequences. (Running on oeis4.)