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A024283 E.g.f. 1/2 * tan(x)^2 (even powers only).
(Formerly N1950)
7
0, 1, 8, 136, 3968, 176896, 11184128, 951878656, 104932671488, 14544442556416, 2475749026562048, 507711943253426176, 123460740095103991808, 35125800801971979943936, 11559592093904798920736768, 4356981378562584648085405696 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Number of cyclically reverse alternating permutations of length 2n+1, cf. A024255. - Vladeta Jovovic, May 20 2007

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]

REFERENCES

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

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..100

FORMULA

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

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

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

a(n) ~ (2*n)! * n * 2^(2*n+3) / Pi^(2*n+2). - Vaclav Kotesovec, Aug 22 2014

EXAMPLE

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

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

MAPLE

A024283 := n -> `if`(n=0, 0, (-1)^(n-1)*2^(2*n+1)*polylog(-2*n-1, -1)); # Peter Luschny, Jun 28 2012

MATHEMATICA

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 *)

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 *)

a[ n_] := If[ n < 1, 0, -2 (-4)^n PolyLog[ -1 - 2 n, -1]] (* Michael Somos, Jan 21 2014 *)

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

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

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 *)

PROG

(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

CROSSREFS

Cf. A009764, A024255.

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

Sequence in context: A036915 A238465 A049211 * A134053 A136472 A145404

Adjacent sequences:  A024280 A024281 A024282 * A024284 A024285 A024286

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Extended and signs tested Mar 15 1997.

STATUS

approved

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Last modified February 25 22:44 EST 2018. Contains 299662 sequences. (Running on oeis4.)