A081042 7th binomial transform of (1,6,0,0,0,0,0,0,...).
1, 13, 133, 1225, 10633, 88837, 722701, 5764801, 45294865, 351652861, 2703691669, 20620693177, 156208812697, 1176509412085, 8816899947037, 65787638066353, 488998835524129, 3622389432086509, 26752509108528805, 197038045347164329
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 (14,-49).
Programs
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Magma
[(6*n+7)*7^(n-1): n in [0..25]]; // Vincenzo Librandi, Aug 06 2013
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Mathematica
CoefficientList[Series[(1 - x)/(1 - 7 x)^2, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 06 2013 *) LinearRecurrence[{14,-49},{1,13},20] (* Harvey P. Dale, Jan 24 2014 *)
Formula
a(n) = 14*a(n-1) - 49*a(n-2) for n>1, a(0)=1, a(1)=13.
a(n) = (6*n+7)*7^(n-1).
a(n) = Sum_{k=0..n} (k+1)*6^k*binomial(n, k).
G.f.: (1-x)/(1-7*x)^2.
E.g.f.: exp(7*x)*(1 + 6*x). - Stefano Spezia, Jan 31 2025