A081039 4th binomial transform of (1,3,0,0,0,0,0,.....).
1, 7, 40, 208, 1024, 4864, 22528, 102400, 458752, 2031616, 8912896, 38797312, 167772160, 721420288, 3087007744, 13153337344, 55834574848, 236223201280, 996432412672, 4191888080896, 17592186044416, 73667279060992
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..300
- Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 67.
- Index entries for linear recurrences with constant coefficients, signature (8,-16).
Programs
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Magma
[(3*n+4)*4^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
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Mathematica
CoefficientList[Series[(1 - x)/(1 - 4 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *) LinearRecurrence[{8,-16},{1,7},30] (* Harvey P. Dale, Dec 13 2015 *) -
PARI
a(n)=(3*n+4)*4^(n-1) \\ Charles R Greathouse IV, Oct 07 2015
Formula
a(n) = 8*a(n-1) -16*a(n-2) with n>1, a(0)=1, a(1)=7.
a(n) = (3*n+4)*4^(n-1).
a(n) = Sum_{k=0..n} (k+1)*3^k*binomial(n, k).
G.f.: (1-x)/(1-4*x)^2.
E.g.f.: exp(4*x)*(1 + 3*x). - Stefano Spezia, Jan 31 2025