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

0, 1, 8, 136, 3968, 176896, 11184128, 951878656, 104932671488, 14544442556416, 2475749026562048, 507711943253426176, 123460740095103991808, 35125800801971979943936, 11559592093904798920736768, 4356981378562584648085405696, 1864703851860264785548754812928

Offset: 0

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

Keywords

• nonn
• easy

Comments

Number of cyclically reverse alternating permutations of length 2n+2, cf. A024255. - Vladeta Jovovic, May 20 2007 [Comment corrected by Fausto A. C. Cariboni, Sep 02 2020]

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]

Examples

			(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 + ...
		

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
• Beáta Bényi, Miguel Méndez, José L. Ramírez and Tanay Wakhare, Restricted r-Stirling Numbers and their Combinatorial Applications, arXiv:1811.12897 [math.CO], 2018.

Crossrefs

Cf. A009764, A024255.

Cf. A000182, A102573. A diagonal of A059419.

Programs

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, 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-François Alcover, Feb 05 2016 *)

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

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

a(n) = (-1)^(n-1)*2^(2*n+1)*PolyLog(-2*n-1,-1) for n >= 1. - Peter Luschny, Jun 28 2012

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

a(n) = (4^(n+1)-1)*Gamma(2*(n+1))*zeta(2*(n+1))/Pi^(2*(n+1)) for n >= 1. - Jean-François Alcover, Feb 05 2016

From Peter Bala, Nov 16 2020: (Start)

a(n) = (1/2)*A000182(n+1) for n >= 1.

Conjectural o.g.f.: x/(1 + x - 9*x/(1 - 8*x/(1 + x - 25*x/(1 - 24*x/(1 + x - ... - (2*n+1)^2*x/(1 - 4*n*(n+1)*x/(1 + x - ... ))))))). (End)

a(n) = (-1)^(n-1)*PolyLog(-2*n - 1, i) for n >= 1. - Peter Luschny, Aug 12 2021

Extensions

Extended and signs tested Mar 15 1997.