OFFSET
0,3
COMMENTS
Column 2 of triangle in A331432.
REFERENCES
J. Ser, Les Calculs Formels des Séries de Factorielles. Gauthier-Villars, Paris, 1933, p. 93.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages)
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).
FORMULA
a(n) = 4*a(n-1) - 5*a(n-2) + 5*a(n-4) - 4*a(n-5) + a(n-6) for n>5. - Vincenzo Librandi, Jan 17 2020
From Bruno Berselli, Jan 17 2020:
a(n) = (n*(n + 3)*(n^2 + 3*n - 2) + 4*(-1)^n - 4)/8. Therefore:
a(n) = n*(n + 3)*(n^2 + 3*n - 2)/8 if n is even,
a(n) = n*(n + 3)*(n^2 + 3*n - 2)/8 - 1 if n is odd. (End)
E.g.f.: (1/8)*(4*exp(-x) + (-4 + 8*x + 32*x^2 + 12*x^3 + x^4)*exp(x)). - G. C. Greubel, Mar 22 2022
MATHEMATICA
CoefficientList[Series[x^2(10-5x+x^2)/((1-x)^4(1-x^2)), {x, 0, 50}], x] (* Vincenzo Librandi, Jan 17 2020 *)
Table[(n(n+3)(n^2+3n-2) +4(-1)^n -4)/8, {n, 0, 50}] (* Bruno Berselli, Jan 17 2020 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 60); [0, 0] cat Coefficients(R!( x^2*(10-5*x+x^2)/((1-x)^4*(1-x^2)))); // Vincenzo Librandi, Jan 17 2020
(Sage) [n*(n+3)*(n^2 +3*n -2)/8 - (n%2) for n in (0..50)] # G. C. Greubel, Mar 22 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 16 2020
STATUS
approved