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A024283 E.g.f. 1/2 * tan(x)^2 (even powers only).
(Formerly N1950)
7

%I N1950

%S 0,1,8,136,3968,176896,11184128,951878656,104932671488,14544442556416,

%T 2475749026562048,507711943253426176,123460740095103991808,

%U 35125800801971979943936,11559592093904798920736768,4356981378562584648085405696

%N E.g.f. 1/2 * tan(x)^2 (even powers only).

%C Number of cyclically reverse alternating permutations of length 2n+1, cf. A024255. - _Vladeta Jovovic_, May 20 2007

%C Related to A102573: letting T(q,r) be the coefficient of n^r in the polynomial 2^(q-n)/n times sum(k=0..n binomial(n, k)*k^q), then A024283(x) = sum(k=0..(2*x-1) T(2*x,k)*(-1)^(k+x)*2^k). See Mathematica code below. [_John M. Campbell_, Sep 15 2013]

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 259, T(n,2).

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%H Alois P. Heinz, <a href="/A024283/b024283.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: (1/2)*(tan(z))^2 = (z^2/(1-z^2)/2)*( 1 +2*z^2/( (z^2-1)*(G(0)-2*z^2)), G(k) = (k+2)*(2*k+3)-2*z^2+2*z^2*(k+2)*(2*k+3)/G(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 15 2011

%F O.g.f.: Sum_{n>=1} x^n * Product_{k=1..n} (2*k-1)^2 / (1 + (2*k-1)^2*x). - _Paul D. Hanna_, Feb 01 2013

%F G.f.: x/(Q(0)-x), where Q(k) = 1 + 2*x*(2*k+1)^2 - x*(2*k+3)^2*(1+x*(2*k+1)^2)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 27 2013

%F a(n) ~ (2*n)! * n * 2^(2*n+3) / Pi^(2*n+2). - _Vaclav Kotesovec_, Aug 22 2014

%e (tan x)^2 = x^2 + 2/3*x^4 + 17/45*x^6 + 62/315*x^8 + ...

%e G.f. = x + 8*x^2 + 136*x^3 + 3968*x^4 + 176896*x^5 + 11184128*x^6 + ...

%p A024283 := n -> `if`(n=0,0,(-1)^(n-1)*2^(2*n+1)*polylog(-2*n-1,-1)); # _Peter Luschny_, Jun 28 2012

%t f[n_] := -(-1)^n 2^(2 n + 1) PolyLog[-1 - 2 n, -1]; f[0] = 0; Array[f, 15, 0] (* _Robert G. Wilson v_, Jun 28 2012 *)

%t poly[q_] := 2^(q-n)/n*FunctionExpand[Sum[Binomial[n, k]*k^q, {k, 0, n}]]; T[q_, r_] := First[Take[CoefficientList[poly[q], n], {r+1, r+1}]]; Print[Table[Sum[T[2*x, k]*(-1)^(k+ x)*(2^k), {k, 0, 2*x-1}], {x, 1, 10}]]; (* _John M. Campbell_, Sep 15 2013 *)

%t a[ n_] := If[ n < 1, 0, -2 (-4)^n PolyLog[ -1 - 2 n, -1]] (* _Michael Somos_, Jan 21 2014 *)

%t a[ n_] := If[ n < 1, 0, With[ {k = 2 n + 1}, k! SeriesCoefficient[ Tan[x] / 2, {x, 0, k}]]] (* _Michael Somos_, Jan 21 2014 *)

%t a[ n_] := If[ n < 0, 0, With[ {k = 2 n}, k! SeriesCoefficient[ Tan[x]^2 / 2, {x, 0, k}]]] (* _Michael Somos_, Jan 21 2014 *)

%t a[0] = 0; a[n_] := (4^(n+1)-1)*Gamma[2*(n+1)]*Zeta[2*(n+1)]/Pi^(2*(n+1)); Table[a[n], {n, 0, 15}] (* _Jean-Fran├žois Alcover_, Feb 05 2016 *)

%o (PARI) {a(n)=polcoeff( sum(m=1, n, x^m*prod(k=1, m, (2*k-1)^2/(1+(2*k-1)^2*x +x*O(x^n))) ), n)} \\ _Paul D. Hanna_, Feb 01 2013

%Y Cf. A009764, A024255.

%Y Cf. A009764, A000182, A102573. A diagonal of A059419.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_. This sequence was in the 1973 "Handbook", but was then omitted from the database. Resubmitted by _R. H. Hardin_. Entry revised by _N. J. A. Sloane_, Jun 12 2012

%E Extended and signs tested Mar 15 1997.

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Last modified February 23 23:56 EST 2018. Contains 299595 sequences. (Running on oeis4.)