A081040 5th binomial transform of (1,4,0,0,0,0,...).
1, 9, 65, 425, 2625, 15625, 90625, 515625, 2890625, 16015625, 87890625, 478515625, 2587890625, 13916015625, 74462890625, 396728515625, 2105712890625, 11138916015625, 58746337890625, 308990478515625, 1621246337890625
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 (10,-25).
Programs
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Magma
[(4*n+5)*5^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
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Mathematica
CoefficientList[Series[(1 - x) / (1 - 5 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *) LinearRecurrence[{10,-25},{1,9},30] (* Harvey P. Dale, Jan 10 2021 *) -
PARI
a(n)=(4*n+5)*5^(n-1) \\ Charles R Greathouse IV, Oct 07 2015
Formula
a(n) = 10*a(n-1) - 25*a(n-2), a(0)=1, a(1)=9.
a(n) = (4n+5)*5^(n-1).
a(n) = Sum_{k=0..n} (k+1)*4^k*binomial(n, k).
G.f.: (1-x)/(1-5*x)^2.
E.g.f.: exp(5*x)*(1 + 4*x). - Stefano Spezia, Jan 31 2025