A080957 Expansion of (5 - 9*x + 6*x^2)/(1-x)^4.

5, 11, 20, 34, 55, 85, 126, 180, 249, 335, 440, 566, 715, 889, 1090, 1320, 1581, 1875, 2204, 2570, 2975, 3421, 3910, 4444, 5025, 5655, 6336, 7070, 7859, 8705, 9610, 10576, 11605, 12699, 13860, 15090, 16391, 17765, 19214, 20740, 22345, 24031, 25800

Offset: 0

Author

Paul Barry, Mar 01 2003

Keywords

• easy
• nonn

Comments

Coefficient of x in the polynomial 6*(C(n,0) + C(n+1,1)*x + C(n+2,2)*x*(x-1)/2 + C(n+3,3)*x*(x-1)*(x-2)/6).

Links

• Danny Rorabaugh, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A080956, A117951.

Programs

Magma

[(2*n^3+3*n^2+31*n+30)/6: n in [0..50]]; // Vincenzo Librandi, Sep 07 2015

Mathematica

CoefficientList[Series[(5-9 x +6 x^2)/(1-x)^4, {x, 0, 45}], x] (* Vincenzo Librandi Sep 07 2015 *)
LinearRecurrence[{4,-6,4,-1},{5,11,20,34},50] (* Harvey P. Dale, Dec 23 2018 *)

PARI

Vec((5-9*x+6*x^2)/(1-x)^4 + O(x^60)) \\ Michel Marcus, Sep 06 2015

PARI

a(n)=(2*n^3 + 3*n^2 + 31*n + 30)/6;
vector(40, n, a(n-1)) \\ Altug Alkan, Sep 28 2015

SageMath

def A080957(n): return (2*n^3 +3*n^2 +31*n +30)//6
print([A080957(n) for n in range(51)]) # G. C. Greubel, May 08 2025

Formula

a(n) = 3!*(C(n+1, 1) - C(n+2, 2)/2 + C(n+3, 3)/3) = (2*n^3 + 3*n^2 + 31*n + 30)/6.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4), n>3. - Vincenzo Librandi, Sep 07 2015

a(n+1) = a(n) + A117951(n+1), a(0) = 5. - Altug Alkan, Sep 28 2015

E.g.f.: (1/6)*(30 + 36*x + 9*x^2 + 2*x^3)*exp(x). - G. C. Greubel, May 08 2025