|
%I M3179 N1288 #76 Oct 28 2023 11:24:57
%S 1,3,567,43659,392931,1724574159,2498907956391,1671769422825579,
%T 88417613265912513891,21857510418232875496803,
%U 2296580829004860630685299,3133969138162958884235052785487,6456973729353591041508572318079423
%N Numerators of Hurwitz numbers H_n (coefficients in expansion of Weierstrass P-function).
%C Named after the German mathematician Adolf Hurwitz (1859-1919). - _Amiram Eldar_, Jun 24 2021
%D Franz Lemmermeyer, Reciprocity Laws, Springer-Verlag, 2000; see p. 276.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A002306/b002306.txt">Table of n, a(n) for n = 1..152</a> (first 60 terms from T. D. Noe)
%H L. Carlitz, <a href="http://dx.doi.org/10.1090/S0025-5718-1962-0152678-2">The coefficients of the lemniscate function</a>, Math. Comp., Vol. 16, No. 80 (1962), pp. 475-478.
%H A. Hurwitz, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN252457811_1897&DMDID=DMDLOG_0041&LOGID=LOG_0041&PHYSID=PHYS_0289">Über die Entwicklungskoeffizienten der lemniskatischen Funktionen</a>, Math. Ann., Vol. 51 (1899), pp. 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
%H A. Hurwitz, <a href="/A002306/a002306.pdf">Über die Entwicklungskoeffizienten der lemniskatischen Funktionen</a>, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII. [Annotated scanned copy]
%F Let P be the Weierstrass P-function satisfying P'^2 = 4*P^3 - 4*P. Then P(z) = 1/z^2 + Sum_{n>=1} 2^(4n)*H_n*z^(4n-2)/(4n*(4n-2)!).
%F Sum_{ (r, s) != (0, 0) } 1/(r+si)^(4n) = (2w)^(4n)*H_n/(4n)! where w = 2 * Integral_{0..1} dx/(sqrt(1-x^4)).
%F See PARI line for recurrence.
%e Hurwitz numbers H_1, H_2, ... = 1/10, 3/10, 567/130, 43659/170, 392931/10, ... = A002306/A047817.
%p H := proc(n) local k; option remember; if n = 1 then 1/10 else 3*add((4*k - 1)*(4*n - 4*k - 1)*binomial(4*n, 4*k)*H(k)*H(n - k), k = 1 .. n - 1)/( (2*n - 3)*(16*n^2 - 1)) fi; end; A002306 := n -> numer(H(n)); seq(A002306(n),n=1..15);
%p # Alternative program
%p c := n -> (n*(4*n-2)!/(2^(4*n-2)))*coeff(series(WeierstrassP(z,4,0),z, 4*n+2),z,4*n-2); a := n -> numer(c(n)); seq(a(n), n=1..13); # _Peter Luschny_, Aug 18 2014
%t a[1] = 1/10; a[n_] := a[n] = (3/(2*n - 3)/(16*n^2 - 1))* Sum[(4*k - 1)*(4*n - 4*k - 1)*Binomial[4*n, 4*k]*a[k]* a[n - k], {k, 1, n - 1}]; Numerator[ Table[a[n], {n, 1, 13}]] (* _Jean-François Alcover_, Oct 18 2011, after PARI *)
%t p[z_] := WeierstrassP[z, {4, 0}]; a[n_] := (n*(4*n-2)!/(2^(4*n-2))) * SeriesCoefficient[p[z], {z, 0, 4*n-2}] // Numerator; Array[a, 13] (* _Jean-François Alcover_, Sep 07 2012, updated Oct 22 2016 *)
%t a[ n_] := If[ n < 0, 0, Numerator[ 2^(-4 n) (4 n)! SeriesCoefficient[ 1 - x WeierstrassZeta[ x, {4, 0}], {x, 0, 4 n}]]]; (* _Michael Somos_, Mar 05 2015 *)
%o (PARI) do(lim)=v=vector(lim); v[1]=1/10; for(n=2,lim,v[n]=3/(2*n-3)/(16*n^2-1)*sum(k=1,n-1,(4*k-1)*(4*n-4*k-1)*binomial(4*n,4*k)*v[k]*v[n-k])) \\ Henri Cohen, Mar 18 2002
%Y Denominators give A047817.
%K nonn,easy,nice,frac
%O 1,2
%A _N. J. A. Sloane_
|
