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A142999 a(0) = 0, a(1) = 1; for n>1, a(n+1) = (2*n+1)*a(n) + n^4*a(n-1). 8
0, 1, 3, 31, 460, 12076, 420336, 21114864, 1325949696, 109027627776, 10771080883200, 1316468976307200, 187978181665996800, 31997755234356019200, 6232784237890147123200, 1409976507981835100160000, 359243973790625586216960000, 104259271562188189469245440000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This is the case m = 0 of the general recurrence a(0) = 1, a(1) = 1, a(n+1) = (2*m+1)*(2*n+1)*a(n) + n^4*a(n-1) (we suppress the dependence of a(n) on m), which arises when accelerating the convergence of the series sum {k = 1..inf} (-1)^(k+1)/k^2 for the constant 1/2*zeta(2). For other cases see A143000 (m=1), A143001 (m=2) and A143002 (m=3).

The solution to the general recurrence may be expressed as a sum: a(n) = n!^2*p_m(n)*sum {k = 1..n} (-1)^(k+1)/(k^2*p_m(k-1)*p_m(k)), where p_m(x) := sum {k = 0..m} C(m,k)*C(x,k)*C(x+k,k). Note that the polynomial q_m(x) := sum {k = 0..m} C(m,k)*C(m+k,k)*C(x,k), obtained by interchanging the roles of m and x, may be variously described as the Ehrhart polynomial of the polytope formed from the convex hull of a root system of type A_m, the polynomial that generates the crystal ball sequence for the A_m lattice [Bacher et al.], or the discrete Chebyshev polynomial D_m(N;x) at N = -1 [Gogin & Hirvensalo]. Compare with the comments in A142995.

The first few values are p_0(x) = 1, p_1(x) = x^2 + x + 1, p_2(x) = (x^4 + 2*x^3 + 7*x^2 + 6*x + 4)/4 and p_3(x) = (x^6 + 3*x^5 + 22*x^4 + 39*x^3 + 85*x^2 + 66*x + 36)/36.

The polynomial p_m(x) is the unique polynomial solution of the difference equation (x+1)^2*f(x+1) - x^2*f(x-1) = (2*m+1)*(2x+1)*f(x), normalized so that f(0) = 1. These polynomials have their zeros on the vertical line Re x = -1/2 in the complex plane; that is, the polynomials p_m(x-1), m = 1,2,3,..., satisfy a Riemann hypothesis (adapt the proof of the lemma on p.4 of [BUMP et al.]).

The general recurrence in the first paragraph above has a second solution b(n) = n!^2*p_m(n) with initial conditions b(0) = 1, b(1) = 2*m+1. Hence the behavior of a(n) for large n is given by lim n -> infinity a(n)/b(n) = sum {k = 1..inf} (-1)^(k+1)/(k^2*p_m(k-1)*p_m(k)) = 1/((2*m+1)+ 1^4/(3*(2*m+1)+ 2^4/(5*(2*m+1)+...+ n^4/(((2*n+1)*(2*m+1)+...)))) = 1/2*sum {k = 1..inf} 1/(m+k)^2. The final equality follows from a result of Ramanujan; see [Berndt, Chapter 12, Corollary to Entry 30].

For results of a similar nature for the constants e, log(2), zeta(2) and zeta(3) see A000522, A142979, A142995 and A143003 respectively.

REFERENCES

Bruce C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et polynômes d'Ehrhart associées aux réseaux de racines, Annales de l'Institut Fourier, Tome 49 (1999) no. 3, pp. 727-762.

R. Bacher, P. de la Harpe and B. Venkov, Séries de croissance et séries d'Ehrhart associées aux réseaux de racines, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.

D. Bump, K. Choi, P. Kurlberg and J. Vaaler, A local Riemann hypothesis, I, Math. Zeit. 233, (2000), 1-19.

N. Gogin and M. Hirvensalo, On the generating function of discrete Chebyshev polynomials, Turku Centre for Computer Science Technical Report No. 819, (2007), 1-8.

FORMULA

a(n) = n!^2*Sum_{k = 1..n} (-1)^(k+1)/k^2.

Recurrence: a(0) = 0, a(1) = 1, a(n+1) = (2*n+1)*a(n) + n^4*a(n-1).

The sequence b(n):= n!^2 satisfies the same recurrence with the initial conditions b(0) = 1, b(1) = 1. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(1+ 1^4/(3+ 2^4/(5+ 3^4/(7+...+ (n-1)^4/(2*n-1))))), for n >=2. Lim n -> infinity a(n)/b(n) = 1/(1+ 1^4/(3+ 2^4/(5+ 3^4/(7+...+ n^4/((2*n+1)+...))))) = sum {k = 1..inf} (-1)^(k+1)/k^2 = 1/2*zeta(2).

Sum_{n>=0} a(n) * x^n / (n!)^2 = -polylog(2,-x) / (1 - x). - Ilya Gutkovskiy, Jul 15 2020

MAPLE

a := n -> n!^2*add ((-1)^(k+1)/k^2, k = 1..n): seq(a(n), n = 0..20);

MATHEMATICA

f[k_] := (k^2) (-1)^(k + 1)

t[n_] := Table[f[k], {k, 1, n}]

a[n_] := SymmetricPolynomial[n - 1, t[n]]

Table[a[n], {n, 1, 18}]    (* A142999 signed *)

(* Clark Kimberling, Dec 30 2011 *)

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(2(n-1)+1)a[n-1]+(n-1)^4 a[n-2]}, a, {n, 20}] (* Harvey P. Dale, Apr 26 2014 *)

CROSSREFS

Cf. A000522, A142979, A142995, A143000, A143001, A143002, A143003.

Sequence in context: A061053 A047798 A126346 * A212917 A223993 A342206

Adjacent sequences:  A142996 A142997 A142998 * A143000 A143001 A143002

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Jul 18 2008

EXTENSIONS

a(0)=0 added by Vincenzo Librandi, Apr 27 2014

STATUS

approved

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Last modified April 11 03:01 EDT 2021. Contains 342886 sequences. (Running on oeis4.)