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A163939
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Triangle related to the o.g.f.s. of the right hand columns of A163934 (E(x,m=4,n)).
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7
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1, 6, 4, 35, 60, 10, 225, 690, 325, 20, 1624, 7588, 6762, 1316, 35, 13132, 85288, 120358, 46928, 4508, 56, 118124, 1004736, 2028660, 1298860, 265365, 13896, 84, 1172700, 12529400, 33896400, 31862400, 11077255, 1313610, 39915, 120
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OFFSET
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1,2
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COMMENTS
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The asymptotic expansions of the higher order exponential integral E(x,m=4,n) lead to triangle A163934, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A163934 have a nice structure Gf(p) = W4(z,p)/(1-z)^(2*p+2) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc.. The coefficients of the W4(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. The row sums of this triangle lead to A000457, 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)*binomial(m-k+2,3)* binomial(2*n+2,k)*stirling1(m+n-k+1,m-k+2), for 1<= m <=n.
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EXAMPLE
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The first few W4(z,p) polynomials are:
W4(z,p=1) = 1/(1-z)^4
W4(z,p=2) = (6+4*z)/(1-z)^6
W4(z,p=3) = (35+60*z+10*z^2)/(1-z)^8
W4(z,p=4) = (225+690*z+325*z^2+20*z^3)/(1-z)^10
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MAPLE
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with(combinat): a := proc(n, m): add((-1)^(n+k+1)*((m-k+2)*(m-k+1)*(m-k)/3!)*binomial(2*n+2, k)*stirling1(m+n-k+1, m-k+2), k=0..m-1) end: seq(seq(a(n, m), m=1..n), n=1..8); # Johannes W. Meijer, revised Nov 27 2012
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MATHEMATICA
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Table[Sum[(-1)^(n + k + 1)*Binomial[m - k + 2, 3]*Binomial[2*n + 2, k]*StirlingS1[m + n - k + 1, m - k + 2], {k, 0, m - 1}], {n, 1, 50}, {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)* binomial(m-k+2, 3)* binomial(2*n+2, k)*stirling(m+n-k+1, m-k+2, 1)), ", "))) \\ G. C. Greubel, Aug 13 2017
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CROSSREFS
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A000399 equals the first left hand column.
A000292 equals 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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