A088802 -id:A088802 - cp's OEIS frontend

A123854 Denominators in an asymptotic expansion for the cubic recurrence sequence A123851.

1, 4, 32, 128, 2048, 8192, 65536, 262144, 8388608, 33554432, 268435456, 1073741824, 17179869184, 68719476736, 549755813888, 2199023255552, 140737488355328, 562949953421312, 4503599627370496, 18014398509481984, 288230376151711744, 1152921504606846976

Offset: 0

Petros Hadjicostas and Jonathan Sondow, Oct 15 2006

A cubic analog of the asymptotic expansion A116603 of Somos's quadratic recurrence sequence A052129. Numerators are A123853.
Equals 2^A004134(n); also the denominators in expansion of (1-x)^(-1/4). - Alexander Adamchuk, Oct 27 2006
All terms are powers of 2 and log_2 a(n) = A004134(n) = 3*n - A000120(n). - Alexander Adamchuk, Oct 27 2006 [Edited by Petros Hadjicostas, May 14 2020]
Is this the same sequence as A088802? - N. J. A. Sloane, Mar 21 2007
Almost certainly this is the same as A088802. - Michael Somos, Aug 23 2007
Denominators of Gegenbauer_C(2n,1/4,2). The denominators of Gegenbauer_C(n,1/4,2) give the doubled sequence. - Paul Barry, Apr 21 2009
If the Greubel formula in A088802 and the Luschny formula here are correct (they are the same), the sequence is a duplicate of A088802. - R. J. Mathar, Aug 02 2023

			A123851(n) ~ c^(3^n)*n^(- 1/2)/(1 + 3/(4*n) - 15/(32*n^2) + 113/(128*n^3) - 5397/(2048*n^4) + ...) where c = 1.1563626843322... is the cubic recurrence constant A123852.

Steven R. Finch, Mathematical Constants, Cambridge University Press, Cambridge, 2003, p. 446.

G. C. Greubel, Table of n, a(n) for n = 0..1000
T. M. Apostol, On the Lerch zeta function, Pacific J. Math. 1 (1951), 161-167. [In Eq. (3.7), p. 166, the index in the summation for the Apostol-Bernoulli numbers should start at s = 0, not at s = 1. - _Petros Hadjicostas_, Aug 09 2019]
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, J. Math. Anal. Appl. 332 (2007), 292-314.
Eric Weisstein's World of Mathematics, Somos's Quadratic Recurrence Constant.
Aimin Xu, Asymptotic expansion related to the Generalized Somos Recurrence constant, International Journal of Number Theory 15(10) (2019), 2043-2055. [The author gives recurrences and other formulas for the coefficients of the asymptotic expansion using the Apostol-Bernoulli numbers (see the reference above) and the Bell polynomials. - _Petros Hadjicostas_, Aug 09 2019]

Cf. A052129, A112302, A116603, A123851, A123852, A123853 (numerators).

Cf. A004134, A004130, A000120.

f:=proc(t,x) exp(sum(ln(1+m*x)/t^m,m=1..infinity)); end; for j from 0 to 29 do denom(coeff(series(f(3,x),x=0,30),x,j)); od;
# Alternatively:
A123854 := n -> denom(binomial(1/4,n)):
seq(A123854(n), n=0..25); # Peter Luschny, Apr 07 2016

Denominator[CoefficientList[Series[ 1/Sqrt[Sqrt[1-x]], {x, 0, 25}], x]] (* Robert G. Wilson v, Mar 23 2014 *)

vector(25, n, n--; denominator(binomial(1/4,n)) ) \\ G. C. Greubel, Aug 08 2019

# uses[A000120]
def A123854(n): return 1 << (3*n-A000120(n))
[A123854(n) for n in (0..25)]  # Peter Luschny, Dec 02 2012

From Alexander Adamchuk, Oct 27 2006: (Start)
a(n) = 2^A004134(n).
a(n) = 2^(3n - A000120(n)). (End)
a(n) = denominator(binomial(1/4,n)). - Peter Luschny, Apr 07 2016

A004130 Numerators in expansion of (1-x)^{-1/4}.

Original entry on oeis.org

1, 1, 5, 15, 195, 663, 4641, 16575, 480675, 1762475, 13042315, 48612265, 729183975, 2748462675, 20809788825, 79077197535, 4823709049635, 18443593425075, 141400882925575, 543277076503525, 8366466978154285, 32270658344309385

Offset: 0

N. J. A. Sloane

Numerators in expansion of sqrt(1/sqrt(1-4x)). - Paul Barry, Jul 12 2005

Denominators are in A088802. - Michael Somos, Aug 23 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A004134, A004981, A034255, A034385, A048882, A007696, A000265, A049606.

Table[Numerator[Binomial[-1/4, n] (-1)^n], {n, 0, 20}]

{a(n) = if( n<0, 0, numerator( polcoeff( (1 - x +x*O(x^n))^(-1/4), n ) ) ) } /* Michael Somos, Aug 23 2007 */

a(n) = prod(k=1, n, (4k-3)/k * 2^A007814(k)), proved by Mitch Harris, following a conjecture by Ralf Stephan.
a(n) = 2^(e_2((2n)!)-n)/n! Product[4k+1,{k,0,n-1}], where e_2((2n)!) is the highest power of 2 that divides (2n)! (sequence A005187). - Emanuele Munarini, Jan 25 2011
Numerators in (1-4t)^(-1/4) = 1 + t + (5/2)t^2 + (15/2)t^3 + (195/8)t^4 + (663/8)t^5 + (4641/16)t^6 + (16575/16)t^7 + ... = 1 + t + 5*t^2/2! + 45*t^3/3! + 585*t^4/4! + ... = e.g.f. for the quartic factorials A007696 (cf. A094638). - Tom Copeland, Dec 04 2013

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

A126963 Numerators of sequence defined by f(0)=1, f(1)=5/4; f(n) = ( (6n-1)f(n-1) - (2n-1)f(n-2) )/(4n).

Original entry on oeis.org

1, 5, 43, 177, 2867, 11531, 92479, 370345, 11857475, 47442055, 379582629, 1518418695, 24295375159, 97182800711, 777467420263, 3109879375897, 199032580597603, 796130905791967, 6369049515119561, 25476202478636219, 407619274119811709, 1630477163761481141

Offset: 0

N. J. A. Sloane, Mar 20 2007

G. C. Greubel, Table of n, a(n) for n = 0..500
D. Doster, Problem 1318, Three Term Recurrence, Math. Magazine, 63 (1990), 127-128.

Denominators are in A088802.

List([0..25], n-> NumeratorRat( Sum([0..n], k-> Binomial(2*k,k)/8^k) ));  # G. C. Greubel, Jan 29 2020

[Numerator( &+[Binomial(2*k, k)/8^k: k in [0..n]] ): n in [0..25]]; // G. C. Greubel, Jan 29 2020

seq( numer( add(binomial(2*k, k)/8^k, k=0..n) ), n=0..25); # G. C. Greubel, Jan 29 2020

a[n_] := Sqrt[2](1-(Gamma[1/2+n] Hypergeometric2F1[n,1/2+n,1+n,-1])/(Sqrt[Pi] Gamma[1+n])); Table[Numerator[FullSimplify[a[n]]], {n,20}] (* Gerry Martens, Aug 09 2015 *)
f[n_]:= If[n==0, 1, If[n==1, 5/4, ((6*n-1)*f[n-1]-(2*n-1)*f[n-2])/(4*n)]];
Table[Numerator[f[n]], {n, 0, 25}] (* G. C. Greubel, Jan 29 2020 *)

A126963(n)=numerator(sum(k=0,n,binomial(-1/2,k)/(-2)^k)) \\ f(n)=if(n>1,((6*n-1)*f(n-1)-(2*n-1)*f(n-2))/(4*n),(5/4)^n) yields the same results. - M. F. Hasler, Aug 11 2015

[numerator( sum(binomial(2*k, k)/8^k for k in (0..n)) ) for n in (0..25)] # G. C. Greubel, Jan 29 2020

f(n) = Sum_{k=0..n} binomial(-1/2,k)*(-1/2)^k.
f(n) -> sqrt(2) as n -> oo.
G.f.: (sqrt(-x)*arccsc(1-x)/sqrt(2)-(Pi*i*sqrt(x))/sqrt(2)^3)/x. - Vladimir Kruchinin, Oct 10 2012
a(n) = numerator( Sum_{k=0..n} binomial(2*k, k)/8^k ). - G. C. Greubel, Jan 29 2020

A143503 Numerators in the asymptotic expansion of Gamma(x+1/2)/Gamma(x).

Original entry on oeis.org

1, -1, 1, 5, -21, -399, 869, 39325, -334477, -28717403, 59697183, 8400372435, -34429291905, -7199255611995, 14631594576045, 4251206967062925, -68787420596367165, -26475975382085110035, 53392138323683746235, 26275374869163335461975, -105772979046693606062363

Offset: 1

Eric W. Weisstein, Aug 20 2008

			1/sqrt(x^(-1)) - sqrt(x^(-1))/8 + (x^(-1))^(3/2)/128 + (5*(x^(-1))^(5/2))/1024 - (21*(x^(-1))^(7/2))/32768 + ...

F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's Interesting Series, arXiv:1009.4274 [math-ph], 2010-2011. [Except for the signs, see the unnumbered table on p. 7.]
F. J. Dyson, N. E. Frankel and M. L. Glasser, Lehmer's interesting series, Amer. Math. Monthly, 120 (2013), 116-130. [Except for the signs, see Table 4.]
D. H. Lehmer, Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(7) (1985), 449-457.
Eric Weisstein's World of Mathematics, Gamma Function.

Cf. A061549, A088802 (denominators), A222411, A222412.

H := proc(n) local S, i; S := (x/(exp(x)-1))^(3/2)*exp(x/2);
-pochhammer(1/2,n-1)*coeff(series(S,x,n+2),x,n)*2^(4*n-1-add(i,i= convert(n,base,2))) end:
A143503 := n -> (-1)^irem(n-1,6)*H(n-1);
seq(A143503(n), n=1..16); # Peter Luschny, Apr 05 2014

Numerator[CoefficientList[Series[Gamma[x + 1/2]/Gamma[x]/Sqrt[x], {x, Infinity, 20}], 1/x]] (* Vaclav Kotesovec, Oct 09 2023 *)

More terms from Vaclav Kotesovec, Oct 09 2023

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

A088801 Numerators of coefficients of powers of n^(-1) in the Romanovsky series expansion of the mean of the standard deviation from a normal population.

Original entry on oeis.org

1, -3, -7, -9, 59, 483, -2323, -42801, 923923, 30055311, -170042041, -8639161167, 99976667055, 7336972779615, -42962450319915, -4309733345367105, 203289825295660035, 26751125064470578695, -158415664732997134045, -26488943422458070446915

Offset: 0

Eric W. Weisstein, Oct 16 2003

Asymptotic expansion of Gamma(N/2) / Gamma((N-1)/2) = (N/2)^(1/2) * (c(0) + c(1)/N + c(2)/N^2 + ... ). a(n) = numerator(c(n)). - Michael Somos, Aug 23 2007

			b(N) = 1 - 3/(4N) - 7/(32N^2) - 9/(128N^3) + ...

Eric Weisstein's World of Mathematics, Standard Deviation Distribution

Cf. A088802.

a[ n_] := If[ n < 0, 0, Module[{A = 1}, Do[ A += x^k / (4 k) SeriesCoefficient[ (A /. x -> x / (1 + 2 x))^2 - (A/(1 - x))^2 / (1 + 2 x) + O[x]^(k + 2), k + 1], {k, n}]; Numerator@Coefficient[A, x, n]]]; (* Michael Somos, May 24 2015 *)

{a(n) = my(A); if(n < 0, 0, A = 1 + O(x) ; for( k = 1, n, A = truncate(A) + x^2 * O(x^k); A += x^k/4/k * polcoeff( subst( A, x, x/(1+2*x))^2 - A^2/(1-x)^2/(1+2*x), k+1 ) ); numerator( polcoeff( A, n ) ) ) }; /* Michael Somos, Aug 23 2007 */

a

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

cp's OEIS Frontend

A123854 Denominators in an asymptotic expansion for the cubic recurrence sequence A123851.

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A004130 Numerators in expansion of (1-x)^{-1/4}.

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A006934 A series for Pi.

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A126963 Numerators of sequence defined by f(0)=1, f(1)=5/4; f(n) = ( (6*n-1)*f(n-1) - (2*n-1)*f(n-2) )/(4n).

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A143503 Numerators in the asymptotic expansion of Gamma(x+1/2)/Gamma(x).

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

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A126963 Numerators of sequence defined by f(0)=1, f(1)=5/4; f(n) = ( (6n-1)f(n-1) - (2n-1)f(n-2) )/(4n).