A123854 -id:A123854 - cp's OEIS frontend

A274653 Numerators of coefficients of z^n/n! for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

0, 1, 21, 185, 18655, 307503, 12548151, 305496477, 138343008375, 4464248592375, 323592065474535, 13015087974100485, 2301190559547593805, 110887163426713235625, 11570760017278599886875, 649837647729572203369125, 1250848387902442801195686375, 80233244659365977333374518375

Offset: 0

Wolfdieter Lang, Jul 07 2016

For the denominators see A274654.
The coefficients of z^n for the expansion of F_1(1/2,1/2;z) are A274655(n)/A274656(n).
Fricke's hypergeometric function F_1(a,b;z) = Sum_{n > = 0} f(a,b;n)*z^n/n!, satisfies the recurrence
  f(a,b,n) = ((a+n-1)*(b+n-1)/n)*f(a,b;n-1) + c(a,b;n)*(1/(a+n-1) + 1/ (b+n-1) - 2/n), with c(a,b;n) = [z^n/n!]hypergeometric([a,b],[1],z) = risefac(a,n) * risefac(b,n)/n!, where risefac is the rising factorial (Pochhammer's symbol) and the input is f(a,b;0)= 0. See the Fricke I reference, p. 114.
The hypergeometric function F_1(1/2,1/2;z) appears in the formula for (2/Pi) K'(k) + (1/Pi)*log(k^2/16)*(2/Pi)*K(k) = F_1(1/2,1/2;k^2), where K and sqrt(-1)*K' are the real and imaginary quarter periods, and k is the modulus (k^2 is the parameter) of elliptic functions. See the Fricke I reference p. 465, eq. (11), and also Fricke III, p. 2, eq. (3).
(2/Pi)*K(k) = hypergeometric([1/2,1/2],[1],k^2). For the expansion coefficients see A038534/A056982 and also A274657/A123854.

			The sequence of rationals {r(n)} begins:
0, 1/2, 21/32, 185/128, 18655/4096, 307503/16384, 12548151/131072, 305496477/524288, 138343008375/33554432, 4464248592375/134217728, 323592065474535/1073741824, ....
The expansion of F_1(1/2,1/2;z) begins:
(1/2)*z + (21/32)*z^2/2! + (185/128)*z^3/3! + (18655/4096)*z^4/4! + (307503/16384)*z^5/5! + ..., or
(1/2)*z + (21/64)*z^2 + (185/768)*z^3 + (18655/98304)*z^4 + (102501/655360)*z^5 + ...

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Erster Teil, Springer-Verlag, 2012, p. 465, eq. (11) with p.114, eq. (15).
R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Dritter Teil, Springer-Verlag, 2012, p. 2, eq. (3).

Cf. A038534, A056982, A274654, A274655, A274656, A274657, A123854.

a(n) = numerator(r(n)), with the rationals (in lowest terms) r(n) = [z^n/n!]F_1(1/2,1/2;z), with the hypergeometric function F_1 given by Fricke. The recurrence of the coefficients r(n) = f(1/2,1/2;n) is obtained from the general one given above.
r(n) = ((2*n-1)^2/(4*n))*r(n-1) + 2*c(n)/(n*(2*n-1)), n >= 1, r(0) = 0, with c(n) = c(1/2,1/2;n) = ((2*n)!)^2 / (n!^3*2^(4*n)) (see A274657/A123854).
E.g.f. for r(n) is Fricke's F_1(1/2,1/2;z).

A006934 A series for Pi.

Original entry on oeis.org

1, 1, 21, 671, 180323, 20898423, 7426362705, 1874409467055, 5099063967524835, 2246777786836681835, 2490122296790918386363, 1694873049836486741425113, 5559749161756484280905626951, 5406810236613380495234085140851, 12304442295910538475633061651918089

Offset: 0

Simon Plouffe and N. J. A. Sloane

nonn

Formula (21) in Luke (see ref.): Let y = 4*n+1. Then for n -> oo
Pi ~ 4*(n!)^4*2^(4*n)/(y*((2*n)!)^2)*(sum_{k>=0}((-1)^k*y^(-2*k)* A006934(k)/A123854(k)))^2. (Luke does not reference the sequences in this form.) - Peter Luschny, Mar 23 2014
This might be related to the numerators of eq. (18) in N. Elezovic' "Asymptotic Expansions of Central Binomial...", J. Int. Seq. 17 (2014) # 14.2.1. - R. J. Mathar, Mar 23 2014
Several references give an erroneous value of 1874409465055 instead of a(7) in the formula for pi. - M. F. Hasler, Mar 23 2014

Y. L. Luke, The Special Functions and their Approximation, Vol. 1, Academic Press, NY, 1969, see p. 36.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

J. L. Fields, A note on the asymptotic expansion of a ratio of gamma functions, Proc. Edinburgh Math. Soc. (15), 43-45, 1966.
A. Gil, J. Segura, N. M. Temme, Fast and accurate computation of the Weber parabolic cylinder function W(a,x), IMA J. Num. Anal. 31 (2011), 1194-1216, eq (3.8).
A. Lupas, Re: Pi Calculation ?, on mathforum.org, Feb 15 2003.
C. Mortici, On some accurate estimates of pi, Bull. Math. Anal. Appl. 2(4) (2010) 137-139. (Formula (1.5), same typo as in Luke)
Index entries for sequences related to the number Pi

Cf. A088802, A123854, A220412.

A006934_list := proc(n) local k, f, bp;
bp := proc(n,x) option remember; local k; if n = 0 then 1 else -x*add(binomial(n-1,2*k+1)*bernoulli(2*k+2)/(k+1)*bp(n-2*k-2,x), k=0..n/2-1) fi end:
f := n -> 2^(3*n-add(i, i=convert(n,base,2)));
add(bp(2*k,1/4)*binomial(4*k,2*k)*x^(2*k), k=0..n-1);
seq((-1)^k*f(k)*coeff(%,x,2*k), k=0..n-1) end:
A006934_list(15);  # Peter Luschny, Mar 23 2014
# Second solution, without using Nörlund's generalized Bernoulli polynomials, based on Euler numbers:
A006934_list := proc(n) local a,c,j;
c := n -> 4^n/2^add(i, i=convert(n,base,2));
a := [seq((-4)^j*euler(2*j)/(4*j), j=1..n)];
expand(exp(add(a[j]*x^(-j), j=1..n))); taylor(%, x=infinity, n+2);
subs(x=1/x, convert(%,polynom)): seq(c(iquo(j,2))*coeff(%,x,j), j=0..n) end:
A006934_list(14); # Peter Luschny, Apr 08 2014

A006934List[n_] := Module[{c, a, s, sx}, c[k_] := 4^k/2^Total[ IntegerDigits[k, 2]]; a = Table[(-4)^j EulerE[2j]/(4j), {j, 1, n}]; s[x_] = Series[Exp[Sum[a[[j]] x^(-j), {j, 1, n}]], {x, Infinity, n+2}] // Normal; sx = s[1/x]; Table[c[Quotient[j, 2]] Coefficient[sx, x, j], {j, 0, n}]];
A006934List[14] (* Jean-François Alcover, Jun 02 2019, from second Maple program *)

@CachedFunction
def p(n):
    if n < 2: return 1
    return -add(binomial(n-1,k-1)*bernoulli(k)*p(n-k)/k for k in range(2,n+1,2))/2
def A006934(n): return (-1)^n*p(2*n)*binomial(4*n,2*n)*2^(3*n-sum(n.digits(2)))
[A006934(n) for n in (0..14)]  # Peter Luschny, Mar 24 2014

Let p(n,x) = sum(k=0..n, x^k*A220412(n,k))/A220411(n) then a(n) = (-1)^n*p(n,1/4)*A123854(n)*A001448(n). - Peter Luschny, Mar 23 2014

Pi = lim_{n->oo} 2^{4n+2}/((4n+1)*C(2n,n)^2)*(sum_{k=0..oo} (-1)^k*a(k)/(A123854(k)*(4n+1)^{2k}))^2. - M. F. Hasler, Mar 23 2014

a(7) corrected, a(8)-a(14) from Peter Luschny, Mar 23 2014

A348678 Triangle read by rows, T(n, k) = denominator([x^k] M(n, x)) where M(n,x) are the Mandelbrot-Larsen polynomials defined in A347928.

Original entry on oeis.org

1, 1, 2, 1, 4, 8, 1, 1, 8, 16, 1, 8, 32, 32, 128, 1, 1, 16, 64, 64, 256, 1, 1, 32, 128, 256, 512, 1024, 1, 1, 1, 64, 256, 512, 1024, 2048, 1, 16, 128, 256, 2048, 64, 4096, 4096, 32768, 1, 1, 32, 256, 512, 4096, 1024, 8192, 8192, 65536

Offset: 0

Peter Luschny, Oct 29 2021

			Triangle starts:
[0] 1
[1] 1,  2
[2] 1,  4,   8
[3] 1,  1,   8,  16
[4] 1,  8,  32,  32,  128
[5] 1,  1,  16,  64,   64,  256
[6] 1,  1,  32, 128,  256,  512, 1024
[7] 1,  1,   1,  64,  256,  512, 1024, 2048
[8] 1, 16, 128, 256, 2048,   64, 4096, 4096, 32768
[9] 1,  1,  32, 256,  512, 4096, 1024, 8192,  8192, 65536

Neil J. Calkin, Eunice Y. S. Chan, and Robert M. Corless, Some Facts and Conjectures about Mandelbrot Polynomials, Maple Trans., Vol. 1, No. 1, Article 14037 (July 2021).
Michael Larsen, Multiplicative series, modular forms, and Mandelbrot polynomials, in: Mathematics of Computation 90.327 (Sept. 2020), pp. 345-377. Preprint: arXiv:1908.09974 [math.NT], 2019.

T(n, n) = A046161(n).

Cf. A348679 (numerators), A347928, A088802 & A123854 (central elements).

# Polynomials M are defined in A347928.
T := (n, k) -> denom(coeff(M(n, x), x, k)):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

A364660 Numerators of coefficients in expansion of (1 + x)^(1/4).

Original entry on oeis.org

1, 1, -3, 7, -77, 231, -1463, 4807, -129789, 447051, -3129357, 11094993, -159028233, 574948227, -4188908511, 15359331207, -906200541213, 3358272593907, -25000473754641, 93422822977869, -1401342344668035, 5271716439465465, -39777496770512145, 150462705175415505, -4564035390320936985

Offset: 0

Ilya Gutkovskiy, Aug 01 2023

			(1 + x)^(1/4) = 1 + x/4 - 3*x^2/32 + 7*x^3/128 - 77*x^4/2048 + 231*x^5/8192 - 1463*x^6/65536 + ...
Coefficients are 1, 1/4, -3/32, 7/128, -77/2048, 231/8192, -1463/65536, ...

Denominators are A088802, A123854.

Cf. A002596, A004130, A008545, A067622, A364661.

nmax = 24; CoefficientList[Series[(1 + x)^(1/4), {x, 0, nmax}], x] // Numerator
Table[Binomial[1/4, n], {n, 0, 24}] // Numerator

my(x='x+O('x^30)); apply(numerator, Vec((1 + x)^(1/4))) \\ Michel Marcus, Aug 02 2023

A364661 Numerators of coefficients in expansion of (1 + x)^(3/4).

Original entry on oeis.org

1, 3, -3, 5, -45, 117, -663, 1989, -49725, 160225, -1057485, 3556995, -48612265, 168273225, -1177912575, 4161957765, -237231592605, 851242773465, -6147864475025, 22326455198775, -325966245902115, 1195209568307755, -8801088639357105, 32525762362841475, -964930950097630425

Offset: 0

Ilya Gutkovskiy, Aug 01 2023

			(1 + x)^(3/4) = 1 + 3*x/4 - 3*x^2/32 + 5*x^3/128 - 45*x^4/2048 + 117*x^5/8192 - 663*x^6/65536 + ...
Coefficients are 1, 3/4, -3/32, 5/128, -45/2048, 117/8192, -663/65536, ...

Denominators are A088802, A123854.

Cf. A002596, A067002, A364658, A364660.

nmax = 24; CoefficientList[Series[(1 + x)^(3/4), {x, 0, nmax}], x] // Numerator
Table[Binomial[3/4, n], {n, 0, 24}] // Numerator

my(x='x+O('x^30)); apply(numerator, Vec((1 + x)^(3/4))) \\ Michel Marcus, Aug 02 2023

A274657 Numerators of the coefficients of z^n/n! for the expansion of hypergeometric([1/2,1/2],[1];z).

Original entry on oeis.org

1, 1, 9, 75, 3675, 59535, 2401245, 57972915, 13043905875, 418854310875, 30241281245175, 1212400457192925, 213786613951685775, 10278202593831046875, 1070401384414690453125, 60013837619516978071875, 57673297952355815927071875, 3694483615889146090857721875

Offset: 0

Wolfdieter Lang, Jul 07 2016

The denominators are given in A123854.
The main entry is A038534 (with A056982) where comments and references are given.
The complete elliptic integral of the first kind K = K(k) is (Pi/2)*hypergeometric([1/2,1/2],[1];k^2). This is also the real quarter period K of elliptic functions.

			The first rationals r(n) are: 1, 1/4, 9/32, 75/128, 3675/2048, 59535/8192, 2401245/65536, 57972915/262144, 13043905875/8388608, 418854310875/33554432, 30241281245175/268435456, ...

Cf. A038534/A056982, A123854.

With[{n = 20}, Numerator[CoefficientList[Series[2 EllipticK[x]/Pi, {x, 0, n}], x] Range[0, n]!]] (* Jan Mangaldan, Jan 04 2017 *)
Numerator[Table[Gamma[n + 1/2]^2/(Pi Gamma[n + 1]), {n, 0, 20}]] (* Li Han, Feb 05 2021 *)

a(n) = numerator(r(n)) with the rationals (in lowest terms) r(n) = (risefac(1/2,n)^2)/n! = ((2*n)!^2)/((n!^3)*2^(4*n)), with the rising factorial risefac (Pochhammer symbol).

E.g.f. for r(n) is hypergeometric([1/2,1/2],[1];z).

A292754 Numerators of coefficients in an asymptotic expansion of the Wallis sequence in inverse powers of n.

Original entry on oeis.org

1, -1, 5, -11, 83, -143, 625, -1843, 24323, 61477, -14165, -8084893, 31181719, 1682401061, -3166220215, -251783137859, 3865962456803, 394670372519917, -765052915887545, -98394908192751193, 384080734825119709, 60838795345430052431, -119312155199695296505, -22845758944383820991909

Offset: 0

N. J. A. Sloane, Sep 25 2017

sign

Chao-Ping Chen, Richard B. Paris, On the asymptotic expansions of products related to the Wallis, Weierstrass, and Wilf formulas, Applied Mathematics and Computation 293 (2017) 30-39. See (3.12).

Chao-Ping Chen, Richard B. Paris, On the asymptotic expansions of products related to the Wallis, Weierstrass, and Wilf formulas, arXiv:1511.09217 [math.CA], 2015. See (3.12).

Cf. A088802 or A123854 (denominators).

nu[j_] := (-1)^(j+1) ((4 - 2^(1-j)) BernoulliB[j+1] - (j+1) 2^(-j))/(j*(j + 1)); mu[j_] := mu[j] = If[j == 0, 1, Sum[k nu[k] mu[j-k], {k, j}]/j]; Table[Numerator@mu@n, {n, 0, 23}] (* Giovanni Resta, May 29 2019 *)
Numerator[CoefficientList[Series[16^n/(Pi*(2*n + 1) * Binomial[2*n, n]^2), {n, Infinity, 20}], 1/n]] (* Vaclav Kotesovec, Jun 02 2019 *)

nu(j) = (-1)^(j+1)*((4-2^(1-j))*bernfrac(j+1) - (j+1)*2^(-j))/(j*(j+1));
mu(j) = if (j==0, 1, sum(k=1, j, k*nu(k)*mu(j-k))/j);
a(n) = numerator(mu(n)); \\ Michel Marcus, May 29 2019

See (3.8) and (3.11) in Chen link.

More terms from Michel Marcus, May 29 2019

A344910 T(n, k) = denominator([x^k] [z^n] ((1 - iz)/(1 + iz))^(ix)(1 + z^2)^(-3/4)). Denominators of the coefficients of the symmetric Meixner-Pollaczek polynomials P^(3/4)_{n}(x, Pi/2). Triangle read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 4, 1, 1, 1, 6, 1, 3, 32, 1, 6, 1, 3, 1, 80, 1, 3, 1, 15, 128, 1, 720, 1, 18, 1, 45, 1, 2240, 1, 360, 1, 45, 1, 315, 2048, 1, 6720, 1, 720, 1, 45, 1, 315, 1, 322560, 1, 90720, 1, 1080, 1, 945, 1, 2835, 8192, 1, 1612800, 1, 181440, 1, 5400, 1, 1890, 1, 14175

Offset: 0

Peter Luschny, Jul 08 2021

			Triangle starts:
  [0] 1;
  [1] 1, 1;
  [2] 4, 1, 1;
  [3] 1, 6, 1, 3;
  [4] 32, 1, 6, 1, 3;
  [5] 1, 80, 1, 3, 1, 15;
  [6] 128, 1, 720, 1, 18, 1, 45;
  [7] 1, 2240, 1, 360, 1, 45, 1, 315;
  [8] 2048, 1, 6720, 1, 720, 1, 45, 1, 315;
  [9] 1, 322560, 1, 90720, 1, 1080, 1, 945, 1, 2835.

R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues, Springer, 2010; pp. 213-216.

Cf. A344909 (numerators).
Cf. A088802 and A123854 (denominator(binomial(1/4, n))) for column 0.
Cf. A049606 (numerator(n!/2^n)) for column n.

gf := ((1 - I*z)/(1 + I*z))^(I*x)*(1 + z^2)^(-3/4):
serz := series(gf, z, 22): coeffz := n -> coeff(serz, z, n):
row := n -> seq(denom(coeff(coeffz(n), x, k)), k = 0..n):
seq(row(n), n = 0..10);
# Alternative:
CoeffList := p -> denom(PolynomialTools:-CoefficientList(p, x)):
P := proc(n) option remember; if n = 0 then 1 elif n = 1 then 2*x else
expand((1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))) fi end:
ListTools:-Flatten([seq(CoeffList(P(n)), n = 0..10)]);

ForceSimpl[a_] := Collect[Expand[Simplify[FunctionExpand[a]]], x]
f[n_] := I^n Sum[(-1)^k Binomial[-3/4 + I*x, k] Binomial[-3/4 - I*x, n-k], {k, 0, n}] // ForceSimpl;
row[n_] := CoefficientList[f[n], x] // Denominator;
Table[row[n], {n, 0, 10}] // Flatten

T(n, k) = denominator([x^k] P(n, x), where P(n, x) = i^n*Sum_{k=0..n} (-1)^k* binomial(-3/4 + i*x, k)*binomial(-3/4 - i*x, n - k). The polynomials have the recurrence P(n, x) = (1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))), starting with P(0, x) = 1 and P(1, x) = 2*x.

cp's OEIS Frontend

A274653 Numerators of coefficients of z^n/n! for the expansion of Fricke's hypergeometric function F_1(1/2,1/2;z).

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A006934 A series for Pi.

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A348678 Triangle read by rows, T(n, k) = denominator([x^k] M(n, x)) where M(n,x) are the Mandelbrot-Larsen polynomials defined in A347928.

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Maple

A364660 Numerators of coefficients in expansion of (1 + x)^(1/4).

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Mathematica

PARI

A364661 Numerators of coefficients in expansion of (1 + x)^(3/4).

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Mathematica

PARI

A274657 Numerators of the coefficients of z^n/n! for the expansion of hypergeometric([1/2,1/2],[1];z).

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Mathematica

Formula

A292754 Numerators of coefficients in an asymptotic expansion of the Wallis sequence in inverse powers of n.

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Mathematica

PARI

Formula

Extensions

A344910 T(n, k) = denominator([x^k] [z^n] ((1 - i*z)/(1 + i*z))^(i*x)*(1 + z^2)^(-3/4)). Denominators of the coefficients of the symmetric Meixner-Pollaczek polynomials P^(3/4)_{n}(x, Pi/2). Triangle read by rows, T(n, k) for 0 <= k <= n.

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A344910 T(n, k) = denominator([x^k] [z^n] ((1 - iz)/(1 + iz))^(ix)(1 + z^2)^(-3/4)). Denominators of the coefficients of the symmetric Meixner-Pollaczek polynomials P^(3/4)_{n}(x, Pi/2). Triangle read by rows, T(n, k) for 0 <= k <= n.