A053567 Stirling numbers of first kind, s(n+5, n).

-120, 1764, -13132, 67284, -269325, 902055, -2637558, 6926634, -16669653, 37312275, -78558480, 156952432, -299650806, 549789282, -973941900, 1672280820, -2792167686, 4546047198, -7234669596, 11276842500, -17247104875, 25922927745, -38343278610, 55880640270

Offset: 1

Author

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000

Keywords

• easy
• sign

Comments

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

References

• 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.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• 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].
• G. C. Greubel, A Note on Jain basis functions, arXiv:1612.09385 [math.CA], 2016.
• Index entries for linear recurrences with constant coefficients, signature (-11,-55,-165,-330,-462,-462,-330,-165,-55,-11,-1).

Crossrefs

Next |Stirling1| diagonal A112002, 5th diagonal of A130534.

Programs

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

Maple

A053567 := proc(n) (-1)^(n+1)*combinat[stirling1](n+5,n) ; end proc: # R. J. Mathar, Jun 08 2011

Mathematica

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 *)

PARI

a(n) = (-1)^(n-1)*stirling(n+5, n, 1); \\ Michel Marcus, Aug 29 2017

Sage

[stirling_number1(n,n-5)*(-1)^(n+1) for n in range(6, 26)] # Zerinvary Lajos, May 16 2009

Formula

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.

Extensions

Definition edited by Eric M. Schmidt, Aug 29 2017

Incorrect formula removed by Greg Dresden, Aug 26 2019