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A276289
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Expansion of x*(1 + x)/(1 - 2*x)^3.
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5
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0, 1, 7, 30, 104, 320, 912, 2464, 6400, 16128, 39680, 95744, 227328, 532480, 1232896, 2826240, 6422528, 14483456, 32440320, 72220672, 159907840, 352321536, 772800512, 1688207360, 3674210304, 7969177600, 17230200832, 37144756224, 79859548160, 171261820928, 366414397440
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OFFSET
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0,3
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COMMENTS
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Binomial transform of pentagonal numbers (A000326).
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.
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
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LINKS
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FORMULA
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O.g.f.: x*(1 + x)/(1 - 2*x)^3.
E.g.f.: x*(2 + 3*x)*exp(2*x)/2.
a(n) = 6*a(n-1) - 12*a(n-2) + 8*a(n-3).
a(n) = Sum_{k = 0..n} binomial(n,k)*k*(3*k - 1)/2.
a(n) = 2^(n-3)*n*(3*n + 1).
Sum_{n>=1} 1/a(n) = 8*(-3*2^(1/3)*Hypergeometric2F1(1/3,1/3;4/3;-1) + 3 + log(2)) = 1.1906948190529335181687...
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MAPLE
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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
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MATHEMATICA
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LinearRecurrence[{6, -12, 8}, {0, 1, 7}, 40]
Table[2^(n - 3) n (3 n + 1), {n, 0, 40}]
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PROG
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(PARI) concat(0, Vec(x*(1+x)/(1-2*x)^3 + O(x^40))) \\ Altug Alkan, Aug 27 2016
(Magma) [2^(n-3)*n*(3*n+1): n in [0..40]]; // G. C. Greubel, Jun 02 2019
(Sage) [2^(n-3)*n*(3*n+1) for n in (0..40)] # G. C. Greubel, Jun 02 2019
(GAP) List([0..40], n-> 2^(n-3)*n*(3*n+1)) # G. C. Greubel, Jun 02 2019
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CROSSREFS
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Cf. A001793 (binomial transform of triangular numbers), A001788 (binomial transform of squares), A084899 (binomial transform of heptagonal numbers).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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