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Triangle related to the o.g.f.s. of the right-hand columns of A028421 (E(x,m=2,n)).
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%I #11 May 11 2019 18:37:20

%S 1,1,2,2,10,3,6,52,43,4,24,308,472,136,5,120,2088,4980,2832,369,6,720,

%T 16056,53988,49808,13638,918,7,5040,138528,616212,826160,381370,57540,

%U 2167,8,40320,1327392,7472952,13570336,9351260,2469300,222908,4948,9

%N Triangle related to the o.g.f.s. of the right-hand columns of A028421 (E(x,m=2,n)).

%C 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.

%H G. C. Greubel, <a href="/A163937/b163937.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F 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.

%e The first few W2(z,p) polynomials are

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

%e W2(z,p=2) = (1 + 2*z)/(1-z)^4;

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

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

%p 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

%t 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 *)

%o (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

%Y Row sums equal A001147 (n>=1).

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

%Y A000027 is the first right hand column.

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

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

%K easy,nonn,tabl

%O 1,3

%A _Johannes W. Meijer_, Aug 13 2009