A047817 -id:A047817 - cp's OEIS frontend

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

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

Offset: 1

N. J. A. Sloane

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

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

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).

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]

Denominators give A047817.

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

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 *)

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

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.

A002770 Integers connected with coefficients in expansion of Weierstrass P-function.

Original entry on oeis.org

-1, 5, 253, 39299, 13265939, 8616924013, 9833937781275, 18382040180023477, 53311001020080183933, 229658082900486063068939, 1418085582879166915943461879, 12182969300667152908506740224429, 141998788870155117956738989275999795

Offset: 2

N. J. A. Sloane

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).

Seiichi Manyama, Table of n, a(n) for n = 2..152
L. Carlitz, The coefficients of the lemniscate function, Math. Comp., 16 (1962), 475-478.
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.
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]

Cf. A002172, A002306, A047817.

nmax = 20; H[n_] := (n*(4*n - 2)!/(2^(4*n - 2)))*SeriesCoefficient[ WeierstrassP[z, {4, 0}], {z, 0, 4*n - 2}]; pp = Select[Prime[Range[2 nmax]], Mod[#, 4] == 1&]; Scan[(chi[#] = -Sum[JacobiSymbol[x^3 - x, #], {x, 0, # - 1}])&, pp]; a[n_] := H[n] - 1/2 - Sum[If[Divisible[4 n, p - 1], chi[p]^(4*n/(p - 1))/p, 0], {p, pp}]; Table[a[n], {n, 2, nmax}] (* Jean-François Alcover, Dec 11 2014, updated Oct 22 2016 *)

a(n) = A002306(n) / A047817(n) - 1/2 - sum(chi(p)^(4*n / (p-1))/p) where the sum is over primes p of the form 4k+1 such that p-1 divides 4*n and the numbers chi(p) are given by A002172. The resulting a(n) is an integer despite all the rationals. - Sean A. Irvine, Aug 17 2014

More terms from Pab Ter (pabrlos(AT)yahoo.com), May 07 2004

A274398 Numerators of 1/2 + sum(chi(p)^(4n / (p-1))/p) where the sum is over primes p of the form 4k+1 such that p-1 divides 4n and the numbers chi(p) are given by A002172.

Original entry on oeis.org

1, 13, -83, 649, -59, 2089, -7379, 8829, -410479, 84273, -4091, 2032897, -867947, 951417, -47224023, 2228469, -262139, 19669687769, -1048571, 1461748549, -1500199283, 746586657, -16777211, 747004180629, -6777994779, 7113541809, -13667368865299, 29908738140693

Offset: 1

Seiichi Manyama, Sep 26 2016

sign

			H_1 = 1/10 = 1/2 - 2/5 = 1/10, so a(1) = 1.
H_2 = 3/10 = 1/2 + 2^2/5 - 1 = 13/10 - 1, so a(2) = 13.
H_3 = 567/130 = 1/2 - 2^3/5 + 6/13 + 5 = -83/130 + 5, so a(3) = -83.
H_4 = 43659/170 = 1/2 + 2^4/5 + 2/17 + 253 = 649/170 + 253, so a(4) = 649.

Seiichi Manyama, Table of n, a(n) for n = 1..3239

Cf. A002172, A002306, A002770, A047817.

nmax = 28; H[n_] := (n*(4*n - 2)!/(2^(4*n - 2)))*SeriesCoefficient[ WeierstrassP[z, {4, 0}], {z, 0, 4*n - 2}]; pp = Select[Prime[Range[2 nmax]], Mod[#, 4] == 1 &]; Scan[(chi[#] = -Sum[JacobiSymbol[x^3 - x, #], {x, 0, # - 1}])&, pp]; a[n_] := 1/2 + Sum[If[Divisible[4 n, p - 1], chi[p]^(4*n/(p - 1))/p, 0], {p, pp}] // Numerator; Array[a, nmax] (* Jean-François Alcover, Oct 22 2016 *)

The n-th Hurwitz number is A002306(n)/A047817(n) = a(n)/A047817(n) + A002770(n).

a(n) = A002306(n) - A002770(n) * A047817(n) for n > 1.

cp's OEIS Frontend

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

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A002770 Integers connected with coefficients in expansion of Weierstrass P-function.

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A274398 Numerators of 1/2 + sum(chi(p)^(4n / (p-1))/p) where the sum is over primes p of the form 4k+1 such that p-1 divides 4n and the numbers chi(p) are given by A002172.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A002770 Integers connected with coefficients in expansion of Weierstrass P-function.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A274398 Numerators of 1/2 + sum(chi(p)^(4*n / (p-1))/p) where the sum is over primes p of the form 4k+1 such that p-1 divides 4*n and the numbers chi(p) are given by A002172.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A274398 Numerators of 1/2 + sum(chi(p)^(4n / (p-1))/p) where the sum is over primes p of the form 4k+1 such that p-1 divides 4n and the numbers chi(p) are given by A002172.