A002306 Numerators of Hurwitz numbers H_n (coefficients in expansion of Weierstrass P-function). (Formerly M3179 N1288)

1, 3, 567, 43659, 392931, 1724574159, 2498907956391, 1671769422825579, 88417613265912513891, 21857510418232875496803, 2296580829004860630685299, 3133969138162958884235052785487, 6456973729353591041508572318079423

Offset

1,2

Comments

Named after the German mathematician Adolf Hurwitz (1859-1919). - Amiram Eldar, Jun 24 2021

References

Franz Lemmermeyer, Reciprocity Laws, Springer-Verlag, 2000; see p. 276.

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

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..152 (first 60 terms from T. D. Noe)

L. Carlitz, The coefficients of the lemniscate function, Math. Comp., Vol. 16, No. 80 (1962), pp. 475-478.

A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., Vol. 51 (1899), pp. 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII.

A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII. [Annotated scanned copy]

Formula

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)!).

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)).

See PARI line for recurrence.

Example

Hurwitz numbers H_1, H_2, ... = 1/10, 3/10, 567/130, 43659/170, 392931/10, ... = A002306/A047817.

Maple

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);
# Alternative program
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

Mathematica

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 *)
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 *)
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 *)

Prog

(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

Crossrefs

Denominators give A047817.

Sequence in context: A265459 A201431 A171359 * A087574 A153402 A121043

Adjacent sequences: A002303 A002304 A002305 * A002307 A002308 A002309

Keyword

nonn,easy,nice,frac

Author

N. J. A. Sloane

Status

approved