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%I #19 Sep 08 2022 08:46:17
%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.
%C Coefficients in the hypergeometric series identity 1 - 7*x/(x + 6) + 30*x*(x - 1)/((x + 6)*(x + 8)) - 104*x*(x - 1)*(x - 2)/((x + 6)*(x + 8)*(x + 10)) + ... = 0, valid in the half-plane Re(x) > 0. Cf. A077616 and A084901. - _Peter Bala_, May 30 2019
%H G. C. Greubel, <a href="/A276289/b276289.txt">Table of n, a(n) for n = 0..1000</a>
%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..40); # _Paolo P. Lava_, Mar 27 2019
%t LinearRecurrence[{6, -12, 8}, {0, 1, 7}, 40]
%t Table[2^(n - 3) n (3 n + 1), {n, 0, 40}]
%o (PARI) concat(0, Vec(x*(1+x)/(1-2*x)^3 + O(x^40))) \\ _Altug Alkan_, Aug 27 2016
%o (Magma) [2^(n-3)*n*(3*n+1): n in [0..40]]; // _G. C. Greubel_, Jun 02 2019
%o (Sage) [2^(n-3)*n*(3*n+1) for n in (0..40)] # _G. C. Greubel_, Jun 02 2019
%o (GAP) List([0..40], n-> 2^(n-3)*n*(3*n+1)) # _G. C. Greubel_, Jun 02 2019
%Y Cf. A000326, A084901.
%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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