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Expansion of (1+2*x+3*x^2+4*x^3+5*x^4)/(1-x-x^2-x^3-x^4-x^5).
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%I #37 Sep 08 2022 08:44:47

%S 1,3,7,15,31,57,113,223,439,863,1695,3333,6553,12883,25327,49791,

%T 97887,192441,378329,743775,1462223,2874655,5651423,11110405,21842481,

%U 42941187,84420151,165965647,326279871,641449337,1261056193,2479171199,4873922247,9581878847

%N Expansion of (1+2*x+3*x^2+4*x^3+5*x^4)/(1-x-x^2-x^3-x^4-x^5).

%C Traces of successive powers of pentanacci matrix. - _Artur Jasinski_, Jan 05 2007

%H G. C. Greubel, <a href="/A023424/b023424.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..199 from T. D. Noe)

%H Martin Burtscher, Igor Szczyrba, RafaƂ Szczyrba, <a href="http://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.pdf">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.

%H S. Saito, T. Tanaka, N. Wakabayashi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Saito/saito22.html">Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values </a>, J. Int. Seq. 14 (2011) # 11.2.4, Table 3.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Lucasn-StepNumber.html">Lucas n-Step Number</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,1,1,1).

%F a(n) = n * Sum_{k=1..n} (1/k)*Sum_{r=0..k} binomial(k,r)*Sum_{m=0..r} binomial(r,m) * Sum_{j=0..m} binomial(m,j)*binomial(j,n-m-k-j-r), n>0. - _Vladimir Kruchinin_, Feb 22 2011

%t LinearRecurrence[{1, 1, 1, 1, 1}, {1, 3, 7, 15, 31}, 60] (* _Vladimir Joseph Stephan Orlovsky_, Feb 08 2012 *)

%t CoefficientList[Series[(1+2*x+3*x^2+4*x^3+5*x^4)/(1-x-x^2-x^3-x^4-x^5), {x, 0, 50}], x] (* _G. C. Greubel_, Jan 01 2018 *)

%o (Maxima)

%o a(n):=n*sum(1/k*sum(binomial(k,r)*sum(binomial(r,m)*sum(binomial(m,j)*binomial(j,n-m-k-j-r),j,0,m),m,0,r),r,0,k),k,1,n);

%o (PARI) Vec((1+2*x+3*x^2+4*x^3+5*x^4)/(1-x-x^2-x^3-x^4-x^5)+O(x^100)) \\ _Charles R Greathouse IV_, Feb 24, 2011

%o (Magma) I:=[1,3,7,15,31]; [n le 5 select I[n] else Self(n-1) + Self(n-2) + Self(n-3) + Self(n-4) + Self(n-5): n in [1..30]]; // _G. C. Greubel_, Jan 01 2018

%Y Essentially the same as A074048.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_