A064096 Fifth diagonal of triangle A064094.

1, 14, 67, 190, 413, 766, 1279, 1982, 2905, 4078, 5531, 7294, 9397, 11870, 14743, 18046, 21809, 26062, 30835, 36158, 42061, 48574, 55727, 63550, 72073, 81326, 91339, 102142, 113765, 126238, 139591, 153854, 169057, 185230, 202403, 220606, 239869, 260222, 281695, 304318, 328121, 353134, 379387, 406910

Offset: 0

Author

Wolfdieter Lang, Sep 13 2001

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A001844 (fourth diagonal), A009766, A064094.

Programs

Magma

[(n+1)^3 +2*n^2*(2*n+1): n in [0..50]]; // G. C. Greubel, Nov 07 2024

Mathematica

CoefficientList[Series[(1 + 2*x)*(1 + 8*x + x^2)/(1 - x)^4, {x, 0, 50}], x] (* Wesley Ivan Hurt, Nov 17 2022 *)

SageMath

def A064096(n): return (n+1)^3 +2*n^2*(2*n+1)
[A064096(n) for n in range(51)] # G. C. Greubel, Nov 07 2024

Formula

a(n) = 1+3*n+5*n^2+5*n^3. Fourth row polynomial (n=3) of Catalan triangle A009766.

G.f.: (1+2*x)*(1+8*x+x^2)/(1-x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Nov 17 2022

E.g.f.: (1 + 13*x + 20*x^2 + 5*x^3)*exp(x). - G. C. Greubel, Nov 07 2024

Extensions

More terms added by G. C. Greubel, Nov 07 2024