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A163938
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Triangle related to the o.g.f.s. of the right hand columns of A163932 (E(x, m=3, n)).
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7
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1, 3, 3, 11, 28, 6, 50, 225, 135, 10, 274, 1858, 2092, 486, 15, 1764, 16464, 29148, 13482, 1491, 21, 13068, 158352, 398640, 301220, 70485, 4152, 28, 109584, 1655172, 5552724, 6132780, 2432070, 322971, 10863, 36
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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=3, n) lead to triangle A163932, see A163931 for information on the E(x,m,n). The o.g.f.s. of the right hand columns of triangle A163932 have a nice structure Gf(p) = W3(z,p)/(1-z)^(2*p+1) with p = 1 for the first right hand column, p = 2 for the second right hand column, etc. The coefficients of the W3(z,p) polynomials lead to the triangle given above, n >= 1 and 1 <= m <= n. The row sums of this triangle lead to A001879, 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+1,2) *binomial(2*n+1,k) *stirling1(m+n-k,m-k+1), for 1 <= m <= n.
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EXAMPLE
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The first few W3(z,p) polynomials are:
W3(z,p=1) = 1/(1-z)^3
W3(z,p=2) = (3 + 3*z)/(1-z)^5
W3(z,p=3) = (11 + 28*z + 6*z^2)/(1-z)^7
W3(z,p=4) = (50 + 225*z + 135*z^2 + 10*z^3)/(1-z)^9
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MAPLE
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with(combinat): a := proc(n, m): add((-1)^(n+k+1)*((m-k+1)*(m-k)/2!)*binomial(2*n+1, k)*stirling1(m+n-k, m-k+1), 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 + 1, 2]*Binomial[2*n + 1, k]*StirlingS1[m + n - k, m - k + 1], {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+1, 2)*binomial(2*n+1, k) *stirling(m+n-k, m-k+1, 1)) , ", "))) \\ G. C. Greubel, Aug 13 2017
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CROSSREFS
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A000254 equals the first left hand column.
A000217 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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