A241756 -id:A241756 - cp's OEIS frontend

A277232 Numerators of the partial sums of the cubes of the expansion coefficients of 1/sqrt(1-x).

1, 9, 603, 4949, 2576763, 20864151, 1347632055, 10860010029, 44749069441659, 359788384157147, 23124997294306677, 185685617347012755, 95380005326947177879, 765237422887515344907, 49101291379356533433423, 393721549706169405868509, 12928613856208967961607217787

Offset: 0

Wolfdieter Lang, Nov 11 2016

The denominators seem to coincide with A241756.
These are the partial sums of F. Morley's series Sum_{k>=0} (risefac(m,k)/k!)^3 for m=1/2, where risefac(x,k) = Product_{j=0..k-1} (x+j), and risefac(x,0) = 1. See the Hardy reference, pp. 104, 111.
The Morley formula gives the value of this series for |m| < 2/3 as Gamma(1-3*m/2)/(Gamma(1-m/2)^3)*cos(Pi*m/2). For the present case m=1/2 this value is hypergeometric([1/2,1/2,1/2],[1,1],1) = Pi/Gamma(3/4)^4 given in A091670.

			The rationals r(n) begin: 1, 9/8, 603/512, 4949/4096, 2576763/2097152, 20864151/16777216, 1347632055/1073741824, ...
The limit is given in A091670, approximately 1.3932039296856768591...

G. H. Hardy, Ramanujan, AMS Chelsea Publ., Providence, RI, 2002, p. 104.

Seiichi Manyama, Table of n, a(n) for n = 0..555
F. Morley, On the Series 1 + (p/1)^3 + {p*(p+1)/1.2}^3 + ... , Proc. London Math. Soc. 34 (1902) 397-402, eq. (5), p. 401.
Eric Weisstein's World of Mathematics, Morley's Formula.

Cf. A001147, A000165, A091670, A241756.

a(n) = numerator(r(n)) with the rational r(n) = Sum_{k=0..n} (risefac(1/2,k)/k!)^3 = Sum_{k=0..n} (-1)^k*(binomial(-1/2,k))^3 = Sum_{k=0..n} ((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.

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.

A278144 Decimal expansion of (sqrt(Pi)/(2^(1/4)Gamma(5/8)Gamma(7/8)))^2.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 14 2016

This is the value of hypergeometric([1/4,1/4],[1],-1)^2. See A278143/A241756 for the partial sums of the hypergeometric series hypergeometric([1/2/,1/2,1/2],[1,1],-1) which has this value due to Clausen's formula. See the Hardy reference, p. 106, eq. (7.4.4) where this value is written as (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2.

			The value of the series 1 - (1/2)^3 + (1*3/(2*4))^3 - (1*3*5/(2*4*6))^3 + ... is 0.909172794546929700739778854282651225720527299592205228386414021837...
This is also the value of the series Sum_{n>=0} c(n) with c(n) = Sum_{k=0..n} f(k)*f(n-k), where f(0)=1 and f(k) = (-1)^k*(1*5*9 *** (4*k-3)/(4*8*12 *** (4*k)))^2, k >= 1 (self-convolution of the hypergeometric([1/4,1/4],[1],-1) series).

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 34.
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..5000

Cf. A278143.

pi:=Pi(RealField(110)); (Sqrt(pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2; // Felix Fröhlich, Nov 15 2016

RealDigits[(Pi/Sqrt[2])*(1/(Gamma[5/8]*Gamma[7/8]))^2, 10, 50][[1]] (* G. C. Greubel, Jan 12 2017 *)

(sqrt(Pi)/(2^(1/4)*gamma(5/8)*gamma(7/8)))^2 \\ Felix Fröhlich, Nov 15 2016

Equals hypergeometric([1/2/,1/2,1/2],[1,1],-1) = hypergeometric([1/4,1/4],[1],-1)^2 = Sum_{k>=0} (-1)^k*(risefac(k,1/2)/k!)^3, where risefac(x,m) = Product_{j =0..m-1} (x+j), and risefac(x,0) = 1.
Equals (Gamma(9/8)/(Gamma(5/4)*Gamma(7/8)))^2 = (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2.
Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)^3/64^k. - Amiram Eldar, Jul 04 2023
Equals Gamma(1/8)^4 * (2 - sqrt(2)) / (16 * Pi^2 * Gamma(1/4)^2). - Vaclav Kotesovec, Jul 04 2023

A241755 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, numerators).

Original entry on oeis.org

1, 1, 27, 125, 42875, 250047, 12326391, 78953589, 266468362875, 1795828623875, 98540708249269, 685638992559339, 308969245276647319, 2197380271937921875, 126096314555551359375, 911218671317138401125, 27146115437208870107914875

Offset: 0

Jean-François Alcover, Apr 28 2014

Quoted from SIAM: This sum arises from the calculation of the shift of the frequency of an electromagnetic transverse magnetic wave-mode caused by a small metallic cylinder in a resonant cavity.

			1, 1/8, 27/512, 125/4096, 42875/2097152, 250047/16777216, ...

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. A241756.

a[n_] := Binomial[2*n, n]^2*Binomial[n-1/2, 2*n]*(-1/4)^n; Table[a[n]//Numerator, {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.

A241857 Number of primes p less than prime(n)-1, such that adding prime(n)-1 and p in binary does not require any carry.

Original entry on oeis.org

0, 0, 2, 0, 1, 2, 6, 2, 0, 2, 0, 5, 7, 2, 1, 3, 1, 2, 8, 2, 9, 1, 4, 5, 11, 5, 1, 2, 4, 6, 0, 14, 16, 7, 9, 3, 4, 6, 3, 6, 3, 5, 0, 18, 8, 2, 4, 0, 4, 5, 7, 1, 6, 1, 54, 10, 15, 5, 16, 18, 7, 14, 6, 3, 10, 5, 6, 16, 2, 4, 17, 2, 1, 6, 1, 0, 15, 8, 19, 10, 6, 9

Offset: 1

Vladimir Shevelev, Apr 30 2014

nonn

Or the number of primes less than prime(n)-1, such that

A000120(prime(n)+p-1) = A000120(p) + A000120(prime(n)-1).

			Let n=12. Prime(12)-1=37-1=36. There are only 5 primes less than 36 the adding of which with 36 does not require any carry: 2,3,11,17,19. So a(12)=5.

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A241756, A241758.

def count(x):
    c = 0
    for y in prime_range(x):
        if binomial(y+x-1,y) % 2:
            c += 1
    return c
[count(i) for i in primes_first_n(100)] # - Tom Edgar, May 01 2014

For Mersenne prime(n), a(n)=0; for Fermat prime(n)>3, a(n)= n-1.

More terms from Peter J. C. Moses, Apr 30 2014

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.

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A277232 Numerators of the partial sums of the cubes of the expansion coefficients of 1/sqrt(1-x).

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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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A278144 Decimal expansion of (sqrt(Pi)/(2^(1/4)*Gamma(5/8)*Gamma(7/8)))^2.

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A241755 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, numerators).

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A241857 Number of primes p less than prime(n)-1, such that adding prime(n)-1 and p in binary does not require any carry.

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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.

A278144 Decimal expansion of (sqrt(Pi)/(2^(1/4)Gamma(5/8)Gamma(7/8)))^2.