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Fourth binomial transform of A010686 (period 2: repeat 1,5).
5

%I #30 Jul 23 2024 14:51:17

%S 1,9,57,321,1713,8889,45417,230001,1158753,5820009,29178777,146130081,

%T 731358993,3658920729,18300980937,91524036561,457677578433,

%U 2288560079049,11443316955897,57218134461441,286095321353073,1430490553902969,7152494610927657,35762598578876721

%N Fourth binomial transform of A010686 (period 2: repeat 1,5).

%H Vincenzo Librandi, <a href="/A080961/b080961.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (8,-15).

%F a(n) = 5*a(n-1) + 4*3^(n-1).

%F a(n) = 3*5^n - 2*3^n.

%F G.f.: (1+x)/((1-3*x)*(1-5*x)). - _Klaus Brockhaus_, Nov 26 2009

%F From _Mario C. Enriquez_, Dec 08 2016: (Start)

%F a(n) = A005059(n+1) + A005059(n) = (5^(n+1)+5^n-3^(n+1)-3^n)/2.

%F a(n) = Sum_{k=0..n} A003948(n-k)*3^k = Sum_{k=0..n} (3^k * ceiling(Sum_{v=0..n-k} (5^v - 5^(v-2)))). (End)

%F a(n) = 8*a(n-1) - 15*a(n-2) for n > 1. - _Wesley Ivan Hurt_, Dec 08 2016

%F E.g.f.: exp(3*x)*(3*exp(2*x) - 2). - _Stefano Spezia_, Jul 23 2024

%e G.f. = 1 + 9*x + 57*x^2 + 321*x^3 + 1713*x^4 + 8889*x^5 + 45417*x^6 + 230001*x^7 + ...

%p A080961:=n->3*5^n-2*3^n: seq(A080961(n), n=0..30); # _Wesley Ivan Hurt_, Dec 08 2016

%t CoefficientList[Series[(1 + x)/((1 - 3*x) * (1 - 5*x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Dec 07 2012 *)

%o (Magma) binomtf:=func< V | [ &+[ Binomial(i-1, k-1)*V[k]: k in [1..i] ]: i in [1..#V] ] >;

%o binomtf(binomtf(binomtf(binomtf(&cat[ [1, 5]: n in [1..11] ])))); // _Klaus Brockhaus_, Nov 26 2009

%o (Magma) [3*5^n - 2*3^n: n in [0..30]]; // _Vincenzo Librandi_, Dec 07 2012

%Y Cf. A003948, A005059, A010686, A080960, A080962.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Mar 03 2003

%E Definition corrected, edited by _Klaus Brockhaus_, Nov 26 2009