A064096 - OEIS

%I #15 Nov 07 2024 09:53:35

%S 1,14,67,190,413,766,1279,1982,2905,4078,5531,7294,9397,11870,14743,

%T 18046,21809,26062,30835,36158,42061,48574,55727,63550,72073,81326,

%U 91339,102142,113765,126238,139591,153854,169057,185230,202403,220606,239869,260222,281695,304318,328121,353134,379387,406910

%N Fifth diagonal of triangle A064094.

%H G. C. Greubel, <a href="/A064096/b064096.txt">Table of n, a(n) for n = 0..2500</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

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

%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Wesley Ivan Hurt_, Nov 17 2022

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

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

%o (Magma) [(n+1)^3 +2*n^2*(2*n+1): n in [0..50]]; // _G. C. Greubel_, Nov 07 2024

%o (SageMath)

%o def A064096(n): return (n+1)^3 +2*n^2*(2*n+1)

%o [A064096(n) for n in range(51)] # _G. C. Greubel_, Nov 07 2024

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

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Sep 13 2001

%E More terms added by _G. C. Greubel_, Nov 07 2024