A002306 -id:A002306 - cp's OEIS frontend

A047817 Denominators of Hurwitz numbers H_n (coefficients in expansion of Weierstrass P-function).

10, 10, 130, 170, 10, 130, 290, 170, 4810, 410, 10, 2210, 530, 290, 7930, 170, 10, 351130, 10, 6970, 3770, 890, 10, 214370, 1010, 530, 524290, 557090, 10, 325130, 10, 170, 130, 1370, 290, 5969210, 1490, 10, 1081730, 6970, 10, 3770

Offset: 1

N. J. A. Sloane

Hurwitz showed (see Katz, eqn. 9) that a(n) = product of the prime p = 2 and the primes p of the form 4*k + 1 such that p - 1 divides 4*n. It follows that a(n) is a divisibility sequence, that is, if n | m then a(n) | a(m). - Peter Bala, Jan 08 2014

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

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

T. D. Noe, Table of n, a(n) for n = 1..1000
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]
N. M. Katz, The Congruences of Clausen - von Staudt and Kummer for Bernoulli-Hurwitz Numbers, Mathematische Annalen 216, 1-4 (1975)
Alexei Pantchichkine, Constructions of p-adic L-functions and admissible measures for Hermitian modular forms, Number Theory [math.NT], 2018.
Alexei Pantchichkine, Algebraic differential operators on arithmetic automorphic forms, modular distributions, p-adic interpolation of their critical l values via BGG modules and Hecke algebras, J. Math. Math. Sci., Thang Long Univ. (Viet Nam, 2022) Vol. 1, No. 4, 1-26.
Index to divisibility sequences

For numerators see A002306.

Cf. A160014.

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;
a := n -> denom(H(n));
# Implementation based on Hurwitz's extension of Clausen's theorem:
GenClausen := proc(n) local k,S; map(k->k+1, numtheory[divisors](n));
    S := select(p-> isprime(p) and p mod 4 = 1, %);
    if S <> {} then 2*mul(k,k=S) else NULL fi end:
A047817_list := proc(n) local i; seq(GenClausen(i),i=1..4*n) end;
A047817_list(42); # Peter Luschny, Oct 03 2011
# Implementation based on Weierstrass's P-function:
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 -> denom(c(n)); seq(a(n), n=1..42); # 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}]; Denominator[ Table[a[n], {n, 1, 42}]] (* Jean-François Alcover, Oct 18 2011, after PARI *)
a[ n_] := If[ n < 1, 0, Denominator[ 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.

A082307 Expansion of e.g.f. (1+x)exp(3x)*cosh(x).

Original entry on oeis.org

1, 4, 16, 66, 280, 1208, 5248, 22816, 98944, 427392, 1838080, 7870976, 33568768, 142637056, 604045312, 2550276096, 10737713152, 45097779200, 188979871744, 790276734976, 3298540650496, 13743907405824, 57174629810176

Offset: 0

Paul Barry, Apr 09 2003

Binomial transform of A002306; a(n)=(A082308(n)+A079028(n))/2

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (12,-52,96,-64).

Cf. A082308.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)*Exp(3*x)*Cosh(x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Sep 16 2018

With[{nmax = 50}, CoefficientList[Series[(1 + x)*Exp[3*x]*Cosh[x], {x, 0, nmax}], x]*Range[0, nmax]!] (* G. C. Greubel, Sep 16 2018 *)

x='x+O('x^30); Vec(serlaplace((1+x)*exp(3*x)*cosh(x))) \\ G. C. Greubel, Sep 16 2018

a(n) = ((n+2)*2^(n-1) + (n+4)*4^(n-1))/2.
G.f.: ((1-3x)/(1-4x)^2 + (1-x)/(1-2x)^2)/2.
E.g.f. (1+x)*exp(3*x)*cosh(x).

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

A260779 Coefficients arising from expansion of 1/(2*P(u)) in powers of u, where P is the Weierstrass P-function.

Original entry on oeis.org

1, -72, 48384, -134120448, 1055796166656, -18987644270149632, 676784742282773397504, -43249455805185586718834688, 4599203617006025540525554139136, -768291761151281123722697889747566592, 192565676807771292904270021964021234663424

Offset: 0

N. J. A. Sloane, Aug 02 2015

This is for the lemniscate case where g2=4, g3=0. - Michael Somos, Jul 10 2024

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] See Eq. (16) and Table III.
Tanay Wakhare and Christophe Vignat, Taylor coefficients of the Jacobi theta3(q) function, arXiv:1909.01508 [math.NT], 2019.

Cf. A144849.

A260779 := proc(n)
    option remember;
    if n = 0 then
        1;
    else
        a :=0 ;
        for r from 0 to n-1 do
            s := n-1-r ;
            if s >=0 and s <= n-1 then
            a := a+procname(r)*procname(s) *binomial(4*n,4*r+2) ;
            end if;
        end do:
        a*(-12) ;
    end if;
end proc: # R. J. Mathar, Aug 03 2015

Block[{a}, a[n_] := If[n < 1, Boole[n == 0], Sum[Binomial[4 n, 4 j + 2] a[j] a[n - 1 - j], {j, 0, n - 1}]]; Array[(-12)^#*a[#] &, 11, 0]] (* Michael De Vlieger, Nov 20 2019, after Harvey P. Dale at A144849 *)
a[ n_] := If[n<0, 0, With[{m = 4*n+2}, m!/2*SeriesCoefficient[ 1/WeierstrassP[u, {4, 0}], {u, 0, m}]]]; (* Michael Somos, Jul 10 2024 *)

Hurwitz (Eq. (84)) gives a recurrence.

a(n) = (-12)^n * A144849(n). - R. J. Mathar, Aug 03 2015

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.

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A047817 Denominators of Hurwitz numbers H_n (coefficients in expansion of Weierstrass P-function).

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A082307 Expansion of e.g.f. (1+x)*exp(3*x)*cosh(x).

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

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A260779 Coefficients arising from expansion of 1/(2*P(u)) in powers of u, where P is the Weierstrass P-function.

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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.

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A082307 Expansion of e.g.f. (1+x)exp(3x)*cosh(x).

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.