A081041 6th binomial transform of (1,5,0,0,0,0,0,0,...).
1, 11, 96, 756, 5616, 40176, 279936, 1912896, 12877056, 85660416, 564350976, 3688436736, 23944605696, 154551545856, 992612745216, 6347497291776, 40435908673536, 256721001578496, 1624959306694656, 10257555623510016
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 (12,-36).
Programs
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Magma
[(5*n+6)*6^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
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Mathematica
CoefficientList[Series[(1 - x)/(1 - 6 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *) LinearRecurrence[{12,-36},{1,11},20] (* Harvey P. Dale, Mar 04 2019 *)
Formula
a(n) = 12*a(n-1) - 36*a(n-2) for n>1, a(0)=1, a(1)=9.
a(n) = (5*n+6)*6^(n-1).
a(n) = Sum_{k=0..n} (k+1)*5^k*binomial(n, k).
G.f.: (1-x)/(1-6*x)^2.
E.g.f.: exp(6*x)*(1 + 5*x). - Stefano Spezia, Jan 31 2025