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A053567
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Stirling numbers of first kind, s(n+5, n).
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8
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-120, 1764, -13132, 67284, -269325, 902055, -2637558, 6926634, -16669653, 37312275, -78558480, 156952432, -299650806, 549789282, -973941900, 1672280820, -2792167686, 4546047198, -7234669596, 11276842500, -17247104875, 25922927745, -38343278610, 55880640270
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OFFSET
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1,1
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COMMENTS
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a(n) is equal to (-1)^n times the sum of the products of each distinct grouping of 5 members of the set {1, 2, 3, ..., n + 4}. So, a(1) = (-1)*1*2*3*4*5 = -120, and a(2) = 1*2*3*4*5 + 1*2*3*4*6 + 1*2*3*5*6 + 1*2*4*5*6 + 1*3*4*5*6 + 2*3*4*5*6 = 1764. See comment at A001303. - Greg Dresden, Aug 26 2019
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for linear recurrences with constant coefficients, signature (-11,-55,-165,-330,-462,-462,-330,-165,-55,-11,-1).
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FORMULA
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a(n) = (-1)^n*binomial(n+5, 6)*binomial(n+5, 2)*(3*n^2 + 23*n + 38)/8.
G.f.: -x*(120 - 444*x + 328*x^2 - 52*x^3 + x^4)/(1+x)^11. See row k=4 of triangle A112007 for the coefficients. [G.f. corrected by Georg Fischer, May 19 2019]
E.g.f. with offset 5: exp(x)*(Sum_{m=0..5} A112486(5, m)*(x^(5+m)/(5+m)!).
a(n) = (f(n+4, 5)/10!)*Sum_{m=0..min(5, n-1)} A112486(5, m)*f(10, 5-m)*f(n-1, m)), with the falling factorials f(n, m):=n*(n-1)*, ..., *(n-(m-1)). From the e.g.f.
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MAPLE
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A053567 := proc(n) (-1)^(n+1)*combinat[stirling1](n+5, n) ; end proc: # R. J. Mathar, Jun 08 2011
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MATHEMATICA
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Table[StirlingS1[n+5, n](-1)^(n-1), {n, 30}] (* Harvey P. Dale, Sep 21 2011 *)
(* or *)
CoefficientList[Series[-x*(120 - 444*x + 328*x^2 - 52*x^3 + x^4)/(1+x)^11, {x, 0, 27}], x] (* Georg Fischer, May 19 2019 *)
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PROG
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(Sage) [stirling_number1(n, n-5)*(-1)^(n+1) for n in range(6, 26)] # Zerinvary Lajos, May 16 2009
(Magma) [(-1)^n*Binomial(n+5, 6)*Binomial(n+5, 2)*(3*n^2+23*n+38)/8: n in [1..30]]; // Vincenzo Librandi, Jun 09 2011
(PARI) a(n) = (-1)^(n-1)*stirling(n+5, n, 1); \\ Michel Marcus, Aug 29 2017
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
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EXTENSIONS
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STATUS
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approved
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