A331429 Expansion of x^2*(10-5*x+x^2)/((1-x)^4*(1-x^2)).

0, 0, 10, 35, 91, 189, 351, 594, 946, 1430, 2080, 2925, 4005, 5355, 7021, 9044, 11476, 14364, 17766, 21735, 26335, 31625, 37675, 44550, 52326, 61074, 70876, 81809, 93961, 107415, 122265, 138600, 156520, 176120, 197506, 220779, 246051, 273429, 303031, 334970, 369370, 406350, 446040, 488565, 534061, 582659, 634501

Offset: 0

Author

N. J. A. Sloane, Jan 16 2020

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A331432.

Programs

Magma

R:=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

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 *)

Sage

[n*(n+3)*(n^2 +3*n -2)/8 - (n%2) for n in (0..50)] # G. C. Greubel, Mar 22 2022

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: (Start)

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