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 A099753 a(n) = (2*n+1)^(n+2). 3

%I

%S 1,27,625,16807,531441,19487171,815730721,38443359375,2015993900449,

%T 116490258898219,7355827511386641,504036361936467383,

%U 37252902984619140625,2954312706550833698643,250246473680347348787521,22550116774162743178682911,2154025884392726618070214209

%N a(n) = (2*n+1)^(n+2).

%H G. C. Greubel, <a href="/A099753/b099753.txt">Table of n, a(n) for n = 0..345</a>

%F From _Peter Bala_, Aug 06 2012: (Start)

%F E.g.f.: d^2/dx^2{(2*x/T(2*x))^(3/2)*1/(1 - T(2*x))} = 1 + 27*x + 625*x^2/2! + ..., where T(x) is the tree function sum {n >=1} n^(n-1)*x^n/n! of A000169.

%F For r = 0, 1, 2, ..., the e.g.f. for the sequence (2*n+1)^(n+r) can be expressed in terms of the function U(z) = sum {n >= 0} (2*n+1)^(n-1)*z^(2*n+1)/(2^n*n!). See A214406 for details. In the present case, r = 2, and the resulting e.g.f. is 1/z*U(z)*(1 + 8*U(z)^2 + 3*U(z)^4)/(1 - U(z)^2)^5 taken at z = sqrt(2*x).

%F (End)

%p seq((2*n+1)^(n+2), n=0..30); # _G. C. Greubel_, Sep 03 2019

%t Table[(2*n+1)^(n+2), {n,0,30}] (* _G. C. Greubel_, Sep 03 2019 *)

%o (PARI) vector(30, n, (2*n-1)^(n+1)) \\ _G. C. Greubel_, Sep 03 2019

%o (MAGMA) [(2*n+1)^(n+2): n in [0..30]]; // _G. C. Greubel_, Sep 03 2019

%o (Sage) [(2*n+1)^(n+2) for n in (0..30)] # _G. C. Greubel_, Sep 03 2019

%o (GAP) List([0..30], n-> (2*n+1)^(n+2)); # _G. C. Greubel_, Sep 03 2019

%Y Cf. A000169, A085527, A085528, A214406.

%K nonn,easy

%O 0,2

%A Kari Lajunen (Kari.Lajunen(AT)Welho.com), Nov 11 2004

%E Terms a(13) onward added by _G. C. Greubel_, Sep 03 2019

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Last modified October 1 04:06 EDT 2020. Contains 337441 sequences. (Running on oeis4.)