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A276289 Expansion of x*(1 + x)/(1 - 2*x)^3. 1

%I

%S 0,1,7,30,104,320,912,2464,6400,16128,39680,95744,227328,532480,

%T 1232896,2826240,6422528,14483456,32440320,72220672,159907840,

%U 352321536,772800512,1688207360,3674210304,7969177600,17230200832,37144756224,79859548160,171261820928,366414397440

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

%C Binomial transform of pentagonal numbers (A000326).

%C More generally, the binomial transform of k-gonal numbers is n*Hypergeometric2F1(k/(k-2),1-n;2/(k-2);-1), where Hypergeometric2F1(a,b;c;x) is the hypergeometric function.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8)

%F O.g.f.: x*(1 + x)/(1 - 2*x)^3.

%F E.g.f.: x*(2 + 3*x)*exp(2*x)/2.

%F a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).

%F a(n) = Sum_{k = 0..n} binomial(n,k)*k*(3*k - 1)/2.

%F a(n) = 2^(n-3)*n*(3*n + 1).

%F Sum_{n>=1} 1/a(n) = 8*(-3*2^(1/3)*Hypergeometric2F1(1/3,1/3;4/3;-1) + 3 + log(2)) = 1.1906948190529335181687...

%p a:=series(x*(1+x)/(1-2*x)^3,x=0,31): seq(coeff(a,x,n),n=0..30); # _Paolo P. Lava_, Mar 27 2019

%t LinearRecurrence[{6, -12, 8}, {0, 1, 7}, 29]

%t Table[2^(n - 3) n (3 n + 1), {n, 0, 28}]

%o (PARI) concat(0, Vec(x*(1+x)/(1-2*x)^3 + O(x^99))) \\ _Altug Alkan_, Aug 27 2016

%Y Cf. A000326.

%Y Cf. A001793 (binomial transform of triangular numbers), A001788 (binomial transform of squares), A084899 (binomial transform of heptagonal numbers).

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Aug 27 2016

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Last modified May 25 15:22 EDT 2019. Contains 323572 sequences. (Running on oeis4.)