OFFSET
1,1
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).
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 17 2000
EXTENSIONS
Definition edited by Eric M. Schmidt, Aug 29 2017
Incorrect formula removed by Greg Dresden, Aug 26 2019
STATUS
approved