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A163937
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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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5
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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A000027 is the first right hand column.
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KEYWORD
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AUTHOR
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STATUS
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approved
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