A277232 -id:A277232 - cp's OEIS frontend

A091670 Decimal expansion of Gamma(1/4)^4/(4*Pi^3).

1, 3, 9, 3, 2, 0, 3, 9, 2, 9, 6, 8, 5, 6, 7, 6, 8, 5, 9, 1, 8, 4, 2, 4, 6, 2, 6, 0, 3, 2, 5, 3, 6, 8, 2, 4, 2, 6, 5, 7, 4, 8, 1, 2, 1, 7, 5, 1, 5, 6, 1, 7, 8, 7, 8, 9, 7, 4, 2, 8, 1, 6, 3, 1, 8, 8, 0, 3, 2, 4, 0, 1, 2, 5, 7, 5, 0, 3, 6, 6, 3, 0, 6, 7, 8, 6, 4, 7, 3, 2, 9, 8, 5, 7, 8, 0, 9, 5, 5, 5, 9, 9

Offset: 1

Eric W. Weisstein, Jan 27 2004

Watson's first triple integral.
This is also the value of F. Morley's series from 1902 Sum_{k=0..n} (risefac(k,1/2)/k!)^3 = hypergeometric([1/2,1/2,1/2],[1,1],1) with the rising factorial risefac(n,x). See A277232, also for the Hardy reference and a MathWorld link. - Wolfdieter Lang, Nov 11 2016
This constant is transcendental due to a result of Nesterenko, who proves that Gamma(1/4) is algebraically independent of Pi. - Charles R Greathouse IV, Aug 19 2025

			1.39320392968567685918424626032536824265748121751561787897...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 324.

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 256, 6.1.17 , p. 557, 15.1.26.
M. L. Glasser, I. J. Zucker, Extended Watson integrals for the cubic lattices, Proc. Nat. Acad. Sci., Vol. 74, No. 5 (1977), p. 1800-1801.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.5.1)
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics, Vol. 187, No. 9 (1996), pp. 1319-1348. (English translation)
Tito Piezas III, Watson's triple integrals.
Eric Weisstein's World of Mathematics, Watson's Triple Integrals.
I. J. Zucker, 70+years of the Watson integrals, J. Stat. Phys., Vol. 145, No. 3 (2011), pp. 591-612.
Index entries for transcendental numbers.

Cf. A091671, A091672, A277232, A293238 (inverse), A068466.

SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^4/(4*Pi(R)^3); // G. C. Greubel, Oct 26 2018

Pi/GAMMA(3/4)^4 ; evalf(%) ; # R. J. Mathar, Jun 17 2016

RealDigits[ N[ Gamma[1/4]^4/(4*Pi^3), 102]][[1]] (* Jean-François Alcover, Nov 12 2012, after Eric W. Weisstein *)

1/agm(sqrt(1/2),1)^2 \\ Charles R Greathouse IV, Mar 03 2016

From Joerg Arndt, Nov 27 2010: (Start)
Equals 1/agm(1,sqrt(1/2))^2.
Equals Gamma(1/4)^4 / (4*Pi^3) = Pi / (Gamma(3/4))^4 = hypergeom([1/2,1/2],[1],1/2)^2, see the two Abramowitz - Stegun references. (End)
Equals the square of A175574. Equals A000796/A068465^4. - R. J. Mathar, Jun 17 2016
Equals hypergeom([1/2,1/2,1/2],[1,1],1) - Wolfdieter Lang, Nov 12 2016
Equals Sum_{k>=0} binomial(2*k,k)^3/2^(6*k). - Amiram Eldar, Aug 26 2020

A241756 A finite sum of products of binomial coefficients: Sum_(m=0..n) binomial(-1/4, m)^2*binomial(-1/4, n-m)^2 (C. C. Grosjean's problem, denominators).

Original entry on oeis.org

1, 8, 512, 4096, 2097152, 16777216, 1073741824, 8589934592, 35184372088832, 281474976710656, 18014398509481984, 144115188075855872, 73786976294838206464, 590295810358705651712, 37778931862957161709568, 302231454903657293676544

Offset: 0

Jean-François Alcover, Apr 28 2014

This sequence seems to appear also as denominators of A277232, A277234, and A278143. - Wolfdieter Lang, Nov 16 2016

E. S. Andersen and M. E. Larsen. A finite sum of products of binomial coefficients, Problem 92-18, by C. C. Grosjean, Solution. SIAM Rev. 35 (1993), 645-646.

P. Flajolet, B. Salvy, and Helmut Prodinger, A Finite Sum of Products of Binomial Coefficients, Problem 92-18 by C. C. Grosjean, Solution. SIAM Rev. 35 (1993), 645-646.
C. C. Grosjean, Problem no. 92-18, SIAM Rev. 34 (1992), p. 649.
M. E. Larsen, Summa Summarum, page 114.

Cf. A241755 (numerators), A277232, A277234, A278143.

a[n_] := Binomial[2*n, n]^2*Binomial[n-1/2, 2*n]*(-1/4)^n; Table[a[n]//Denominator, {n, 0, 20}]

GAMMA(3/4)^2 * 4F3(1/4, 1/4, -n, -n; 1, 3/4-n, 3/4-n; 1)/(GAMMA(3/4-n)^2*GAMMA(n+1)^2).
binomial(2n, n)^2*binomial(n-1/2, 2n)*(-1/4)^n.
Conjecture (from sequencedb.net): a(n) = 8^A005187(n). - R. J. Mathar, Jun 30 2021

A277233 Numerators of the partial sums of the squares of the expansion coefficients of 1/sqrt(1-x). Also the numerators of the Landau constants.

Original entry on oeis.org

1, 5, 89, 381, 25609, 106405, 1755841, 7207405, 1886504905, 7693763645, 125233642041, 508710104205, 33014475398641, 133748096600189, 2165115508033649, 8754452051708621, 9054883309760265929, 36559890613417481741, 590105629859261338481, 2379942639329101454549

Offset: 0

Wolfdieter Lang, Nov 12 2016

This is the instance m=1/2 of the partial sums r(m,n) = Sum_{k=0..n} (risefac(m,k)/ k!)^2, where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1.
The limit n -> oo does not exist. It would be hypergeometric([1/2,1/2],[1],z -> 1), which diverges.
The partial sums of the cubes converge for |m| < 2/3. See Morley's series under A277232 (for m=1/2).
a(n)/A056982(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2, where g(n) = (2*n)!/(2^n*n!)^2 = A001790(n)/A046161(n). - Peter Luschny, Sep 27 2019

			The rationals r(n) begin: 1, 5/4, 89/64, 381/256, 25609/16384, 106405/65536, 1755841/1048576, 7207405/4194304, 1886504905/1073741824, 7693763645/4294967296, ...

Seiichi Manyama, Table of n, a(n) for n = 0..831
Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe, Arch. Math. Phys. 21 (1913), 42-50. [Accessible in the USA through the Hathi Trust Digital Library.]
Edmund Landau, Abschätzung der Koeffzientensumme einer Potenzreihe (Zweite Abhandlung), Arch. Math. Phys. 21 (1913), 250-255. [Accessible in the USA through the Hathi Trust Digital Library.]
Cristinel Mortici, Sharp bounds of the Landau constants, Math. Comp. 80 (2011), pp. 1011-1018.
G. N. Watson, The constants of Landau and Lebesgue, Quart. J. Math. Oxford Ser. 1:2 (1930), pp. 310-318.

Denominators are A056982.

Cf. A001803/A046161, A277232/A241756, A001790/A046161, A005187.

a := n -> numer(add(binomial(-1/2, j)^2, j=0..n));
seq(a(n), n=0..19); # Peter Luschny, Sep 26 2019
# Alternatively:
G := proc(x) hypergeom([1/2,1/2], [1], x)/(1-x) end: ser := series(G(x), x, 20):
[seq(coeff(ser,x,n), n=0..19)]: numer(%); # Peter Luschny, Sep 28 2019

Accumulate[CoefficientList[Series[1/Sqrt[1-x],{x,0,20}],x]^2]//Numerator (* Harvey P. Dale, Feb 10 2019 *)
G[x_] := (2 EllipticK[x])/(Pi (1 - x));
CoefficientList[Series[G[x], {x, 0, 19}], x] // Numerator (* Peter Luschny, Sep 28 2019 *)

def A277233(n):
    return sum((A001790(k)*(2^(A005187(n) - A005187(k))))^2 for k in (0..n))
print([A277233(n) for n in (0..19)]) # Peter Luschny, Sep 30 2019

a(n) = numerator(r(n)), with the fractional
r(n) = Sum_{k=0..n} (risefac(1/2,k)/k!)^2;
r(n) = Sum_{k=0..n} (binomial(-1/2,k))^2;
r(n) = Sum_{k=0..n} ((2*k-1)!!/(2*k)!!)^2.
The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
r(n) ~ (log(n+3/4) + EulerGamma + 4*log(2))/Pi. - Peter Luschny, Sep 27 2019
Rational generating function: (2*K(x))/(Pi*(1-x)) where K is the complete elliptic integral of the first kind. - Peter Luschny, Sep 28 2019
a(n) = Sum_{k=0..n}(A001790(k)*(2^(A005187(n) - A005187(k))))^2. - Peter Luschny, Sep 30 2019

A278143 Numerators of partial sums of a hypergeometric series with value Pi/(sqrt(2)(Gamma(5/8)Gamma(7/8))^2) = A278144.

Original entry on oeis.org

1, 7, 475, 3675, 1924475, 15145753, 981654583, 7774283075, 32109931838075, 255083626080725, 16423892777415669, 130705503226766013, 67230186897380845975, 535644114907108845925, 34407319668610517498575, 274347338677567001587475

Offset: 0

Wolfdieter Lang, Nov 14 2016

The denominators appear to be given in A241756.
The series is 1 - (1/2)^3 + (1*3/2*4)^3 -+ ... = Sum_{k>=0} (-1)^k*(risefac(1/2,k)/ k!)^3 = hypergeometric([1/2,1/2,1/2],[1,1],-1), where risefac(x,k) = Product_{j =0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference p. 106.
Due to Clausen's formula given in eq. (7.4.5) this is (hypergeometric([1/2,1/2],[1],-1))^2. Hardy's result in eq. (7.4.4) is (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2 which can be rewritten as (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2. See the Abramowitz-Stegun reference p. 557, 15.1.21 and p. 256, 6.1.18.
This series is the alternating sum version of Morley's series for m=1/2. See A277232. Hence the present sequence gives the numerators of the partial sums of the cubes of the expansion coefficients of 1/sqrt(1+x).

			The rationals begin: 1, 7/8, 475/512, 3675/4096, 1924475/2097152, 15145753/16777216, 981654583/1073741824, 7774283075/8589934592, ... .
The limit r(n), for n -> oo is Pi/(sqrt(2)*(Gamma(5/8)*Gamma(7/8))^2) = 0.90917563087572... given in A278144.

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 106, eq. (7.4.4).

G. C. Greubel, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 557, (15.1.21).

Cf. A241756, A277232, A278144.

Table[Numerator@ Sum[(-1)^k (Pochhammer[1/2, k]/k!)^3, {k, 0, n}], {n, 0, 15}] (* or *)
Table[Numerator@ Sum[Binomial[-1/2, k]^3, {k, 0, n}], {n, 0, 15}] (* or *)
Table[Numerator@ Sum[(-1)^k*((2 k - 1)!!/(2 k)!!)^3, {k, 0, n}], {n, 0, 15}] (* Michael De Vlieger, Nov 15 2016 *)

for(n=0,25, print1(numerator(sum(k=0,n, binomial(-1/2,k)^3)), ", ")) \\ G. C. Greubel, Feb 06 2017

a(n) = numerator(r(n)) with the rational r(n) = Sum_{k=0..n} (-1)^k (risefac(1/2,k)/k!)^3 = Sum_{k=0..n} (binomial(-1/2,k))^3 = Sum_{k=0..n} (-1)^k*((2*k-1)!!/(2*k)!!)^3. The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.

A277235 Decimal expansion of 2/(Gamma(3/4))^4.

Original entry on oeis.org

8, 8, 6, 9, 4, 1, 1, 6, 8, 5, 7, 8, 1, 1, 5, 4, 0, 5, 4, 1, 1, 5, 2, 5, 3, 6, 1, 3, 5, 4, 5, 2, 1, 5, 3, 8, 6, 8, 6, 4, 9, 9, 9, 1, 9, 6, 4, 2, 5, 9, 8, 3, 4, 8, 3, 0, 9, 8, 6, 0, 9, 8, 9, 8, 1, 3, 1, 7, 8, 2, 5, 5, 9, 4, 8, 1, 9, 2, 7, 9, 7, 0, 6, 9, 1, 5, 2, 6, 4, 7, 7, 9, 4, 9, 8, 1, 2, 1

Offset: 0

Wolfdieter Lang, Nov 13 2016

This is the value of one of Ramanujan's series: 1 - 5*(1/2)^5 + 9*(1*3/(2*4))^5 -13*(1*3*5/(2*4*6))^5 + - ... . See the Hardy reference p.7. eq. (1.4) and pp. 105-106. For the partial sums see A278140.

The proof of Hardy and Whipple mentioned in the Hardy reference reduces this series to (2/Pi)*Morley's series (for m=1/2). For this series see A277232 and A091670.

			2/Gamma(3/4)^4 = 0.88694116857811540541152...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A014549, A060294, A074799, A074800, A088538, A091670, A263809, A277232, A278140.

SetDefaultRealField(RealField(100)); 2/(Gamma(3/4))^4; // G. C. Greubel, Oct 26 2018

RealDigits[2/(Gamma[3/4])^4, 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)

2/gamma(3/4)^4 \\ Michel Marcus, Nov 13 2016

Equals Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^5 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^5. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
Equals A060294 * A091670.
For (1+4*k)*((2*k-1)!!/(2*k)!!)^5 see A074799(k) / A074800(k).
From Amiram Eldar, Jul 13 2023: (Start)
Equals (Gamma(1/4)/Pi)^4/2.
Equals A088538 * A014549^2.
Equals A263809/Pi. (End)

A277234 Numerators of partial sums of a Ramanujan series converging to 2/Pi = A060294.

Original entry on oeis.org

1, 3, 435, 1855, 1678635, 8178093, 831557727, 4362807735, 26663516457435, 146862472576105, 13439367283090749, 76661183599555737, 54390019021528255975, 318658997759516188425, 27581665786275463543575, 165068987339858265879975, 7173478080571052213369487675

Offset: 0

Wolfdieter Lang, Nov 13 2016

The denominators seem to be A241756.

One of Ramanujan's series is 1 - 5*(1/2)^3 + 9*(1*3/(2*4))^3 - 13*(1*3*5/(2*4*6))^3 +- ... = Sum_{k>=0} (-1)^k*(1+4*k)*(risefac(1/2,k)/k!)^3 where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, p. 7, eq. (1.2) and p. 105, eq. (7.4.2) for s=1/2. The limit is Buffon's constant 2/Pi given in A060294.

			The rationals r(n) begin: 1, 3/8, 435/512, 1855/4096, 1678635/2097152, 8178093/16777216, 831557727/1073741824, 4362807735/8589934592, ...
The limit r(n), n -> oo, is 2/Pi = 0.6366197723... given in A060294.

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105.

Cf. A060294, A241756, A277232.

a(n) = numerator(r(n)), with the rationals r(n) = Sum_{k=0..n} (-1)^k*(1+4*k)*(risefac(1/2,k)/k!)^3 = Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^3 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^3. The rising factorial has been defined in a comment above. The double factorials are given in A001147 and A000165 with (-1)!! := 1.

cp's OEIS Frontend

A091670 Decimal expansion of Gamma(1/4)^4/(4*Pi^3).

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A241756 A finite sum of products of binomial coefficients: Sum_(m=0..n) binomial(-1/4, m)^2*binomial(-1/4, n-m)^2 (C. C. Grosjean's problem, denominators).

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A277233 Numerators of the partial sums of the squares of the expansion coefficients of 1/sqrt(1-x). Also the numerators of the Landau constants.

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A278143 Numerators of partial sums of a hypergeometric series with value Pi/(sqrt(2)*(Gamma(5/8)*Gamma(7/8))^2) = A278144.

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A277235 Decimal expansion of 2/(Gamma(3/4))^4.

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A277234 Numerators of partial sums of a Ramanujan series converging to 2/Pi = A060294.

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A278143 Numerators of partial sums of a hypergeometric series with value Pi/(sqrt(2)(Gamma(5/8)Gamma(7/8))^2) = A278144.