A004981 -id:A004981 - cp's OEIS frontend

A208890 a(n) = A000984(n)A004981(n), the term-wise product of the coefficients in (1-4x)^(-1/2) and (1-8*x)^(-1/4).

1, 4, 60, 1200, 27300, 668304, 17153136, 455083200, 12372574500, 342766138000, 9638583800560, 274341178587840, 7887308884400400, 228685287180840000, 6678543795015960000, 196260140322869011200, 5798873833602270315300, 172160337343624495866000

Offset: 0

Paul D. Hanna, Mar 04 2012

nonn

The sequences A000984 and A004981 are related by the aesthetic identity:

Sum_{n>=0} A000984(n)^3 *x^n = ( Sum_{n>=0} A004981(n)^2 *x^n )^2.

			G.f.: A(x) = 1 + 4*x + 60*x^2 + 1200*x^3 + 27300*x^4 + 668304*x^5 +...
The terms are the term-wise products of the sequences:
A000984 = [1, 2, 6, 20, 70, 252, 924, 3432, 12870, 48620, ...],
A004981 = [1, 2, 10, 60, 390, 2652, 18564, 132600, 961350, ...].
Related sequences:
A^2: [1, 8, 136, 2880, 67800, 1699008, 44368704, 1193107968, ...],
A^4: [1, 16, 336, 7936, 200176, 5266176, 142657536, 3948773376, ...],
A^8: [1, 32, 928, 26624, 767200, 22270976, 651331072, 19178651648, ...].

Cf. A000984, A004981, A181418.

{A000984(n)=polcoeff((1-4*x +x*O(x^n))^(-1/2),n)}
{A004981(n)=polcoeff((1-8*x +x*O(x^n))^(-1/4),n)}
{a(n)=A000984(n)*A004981(n)}
for(n=0,20,print1(a(n),", "))

A007696 Quartic (or 4-fold) factorial numbers: a(n) = Product_{k = 0..n-1} (4*k + 1).

Original entry on oeis.org

1, 1, 5, 45, 585, 9945, 208845, 5221125, 151412625, 4996616625, 184874815125, 7579867420125, 341094033905625, 16713607661375625, 885821206052908125, 50491808745015763125, 3080000333445961550625, 200200021673987500790625, 13813801495505137554553125

Offset: 0

N. J. A. Sloane

a(n), n >= 1, enumerates increasing quintic (5-ary) trees. See David Callan's comment on A007559 (number of increasing quarterny trees).

Hankel transform is A169619. - Paul Barry, Dec 03 2009

			G.f. = 1 + x + 5*x^2 + 45*x^3 + 585*x^4 + 9945*x^5 + 208845*x^6 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq. 3 (2000), Article 00.2.4.
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations, (m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Integer Seq. 13 (2010), Article 10.6.7; see page 39.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, J. Integer Seq. 9 (2006), Article 06.1.1.

Cf. A001147, A001813, A004981, A007559, A008545, A034255, A047053, A051142, A264781.

a(n) = A049029(n, 1) for n >= 1 (first column of triangle).

a:=[1,1];; for n in [3..20] do a[n]:=(4*(n-1)-7)*(a[n-1]+4*a[n-2]); od; a; # G. C. Greubel, Aug 15 2019

[n le 2 select 1 else (4*(n-1)-7)*(Self(n-1) + 4*Self(n-2)): n in [1..20]]; // G. C. Greubel, Aug 15 2019

x:='x'; G(x):=(1-4*x)^(-1/4): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: seq(eval(f[n],x=0),n=0..17);# Zerinvary Lajos, Apr 03 2009
A007696 := n -> mul(k, k = select(k-> k mod 4 = 1, [$ 1 .. 4*n])): seq(A007696(n), n=0..17); # Peter Luschny, Jun 23 2011

a[ n_]:= Pochhammer[ 1/4, n] 4^n; (* Michael Somos, Jan 17 2014 *)
a[ n_]:= If[n < 0, 1 / Product[ -k, {k, 3, -4n-1, 4}], Product[ k, {k, 1, 4n-3, 4}]]; (* Michael Somos, Jan 17 2014 *)
Range[0, 19]! CoefficientList[Series[((1-4x)^(-1/4)), {x, 0, 19}], x] (* Vincenzo Librandi, Oct 08 2015 *)

A007696(n):=prod(4*k+1,k,0,n-1)$
makelist(A007696(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

{a(n) = if( n<0, 1 / prod(k=1, -n, 1 - 4*k), prod(k=1, n, 4*k - 3))}; /* Michael Somos, Jan 17 2014 */

[4^n*rising_factorial(1/4, n) for n in (0..20)] # G. C. Greubel, Aug 15 2019

E.g.f.: (1 - 4*x)^(-1/4).
a(n) ~ 2^(1/2) * Pi^(1/2) * Gamma(1/4)^(-1) * n^(-1/4) * 2^(2*n) * e^(-n) * n^n * (1 - 1/96 * n^(-1) - ...). - Joe Keane (jgk(AT)jgk.org), Nov 23 2001 [corrected by Vaclav Kotesovec, Jul 19 2025]
a(n) = Sum_{k = 0..n} (-4)^(n-k) * A048994(n, k). - Philippe Deléham, Oct 29 2005
G.f.: 1/(1 - x/(1 - 4*x/(1 - 5*x/(1 - 8*x/(1 - 9*x/(1 - 12*x/(1 - 13*x/(1 - .../(1 - A042948(n+1)*x/(1 -... (continued fraction). - Paul Barry, Dec 03 2009
a(n) = (-3)^n * Sum_{k = 0..n} (4/3)^k * s(n+1, n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/T(0), where T(k) =  1 - x * (4*k + 1)/(1 - x * (4*k + 4)/T(k+1)) (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013
G.f.: 1 + x/Q(0), where Q(k) = 1 + x + 2*(2*k - 1)*x - 4*x*(k+1)/Q(k+1) (continued fraction). - Sergei N. Gladkovskii, May 03 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x * (4*k + 1)/(x * (4*k + 1) + 1/G(k+1))) (continued fraction). - Sergei N. Gladkovskii, Jun 04 2013
0 = a(n) * (4*a(n+1) - a(n+2)) + a(n+1) * a(n+1) for all n in Z. - Michael Somos, Jan 17 2014
a(-n) = (-1)^n / A008545(n). - Michael Somos, Jan 17 2014
Let T(x) = 1/(1 - 3*x)^(1/3) be the e.g.f. for the sequence of triple factorial numbers A007559. Then the e.g.f. A(x) for the quartic factorial numbers satisfies T(Integral_{t = 0..x} A(t) dt) = A(x). (Cf. A007559 and A008548.) - Peter Bala, Jan 02 2015
O.g.f.: hypergeom([1, 1/4], [], 4*x). - Peter Luschny, Oct 08 2015
a(n) = A264781(4*n+1, n). - Alois P. Heinz, Nov 24 2015
a(n) = 4^n * Gamma(n + 1/4)/Gamma(1/4). - Artur Jasinski, Aug 23 2016
D-finite with recurrence: a(n) +(-4*n+3)*a(n-1)=0, n>=1. - R. J. Mathar, Feb 14 2020
Sum_{n>=0} 1/a(n) = 1 + exp(1/4)*(Gamma(1/4) - Gamma(1/4, 1/4))/(2*sqrt(2)). - Amiram Eldar, Dec 18 2022

Better description from Wolfdieter Lang, Dec 11 1999

A059304 a(n) = 2^n * (2*n)! / (n!)^2.

Original entry on oeis.org

1, 4, 24, 160, 1120, 8064, 59136, 439296, 3294720, 24893440, 189190144, 1444724736, 11076222976, 85201715200, 657270374400, 5082890895360, 39392404439040, 305870434467840, 2378992268083200, 18531097667174400

Offset: 0

Henry Bottomley, Jan 25 2001

Number of lattice paths from (0,0) to (n,n) using steps (0,1), and two kinds of steps (1,0). - Joerg Arndt, Jul 01 2011
The convolution square root of this sequence is A004981. - T. D. Noe, Jun 11 2002
Also main diagonal of array: T(i,1)=2^(i-1), T(1,j)=1, T(i,j) = T(i,j-1) + 2*T(i-1,j). - Benoit Cloitre, Feb 26 2003
The Hankel transform (see A001906 for definition) of this sequence with interpolated zeros(1, 0, 4, 0, 24, 0, 160, 0, 1120, ...) = is A036442: 1, 4, 32, 512, 16384, ... . - Philippe Deléham, Jul 03 2005
The Hankel transform of this sequence gives A103488. - Philippe Deléham, Dec 02 2007
Equals the central column of the triangle A038207. - Zerinvary Lajos, Dec 08 2007
Equals number of permutations whose reverse shares the same recording tableau in the Robinson-Schensted correspondence with n=(k-1)/2 for k odd. - Dang-Son Nguyen, Jul 02 2024
Number of ternary strings of length 2*n that have the same number of 0's as the combined number of 1's and 2's. For example, a(2)=24 since the strings of length 4 are the 6 permutations of 0011, the 12 permutations of 0012, and the 6 permutations of 0022. - Enrique Navarrete, Jul 30 2025

Harry J. Smith, Table of n, a(n) for n = 0..200
Paul Barry and Arnauld Mesinga Mwafise, Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.
Hacène Belbachir and Abdelghani Mehdaoui, Recurrence relation associated with the sums of square binomial coefficients, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
H. J. Brothers, Pascal's Prism: Supplementary Material.
Tucker J. Ervin, Blake Jackson, Jay Lane, Kyungyong Lee, Son Dang Nguyen, Jack O'Donohue and Michael Vaughan, Permutations whose Reverse Shares the Same Recording Tableau in the Robinson-Schensted Correspondence, Séminaire Lotharingien de Combinatoire 86 (2022), Article B86a.
Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Diagonal of A013609.
Column k=0 of A067001.
Cf. A000079, A000984, A001906, A036442, A038207, A103488.

[2^n*Binomial(2*n,n): n in [0..25]]; // Vincenzo Librandi, Oct 08 2015

seq(binomial(2*n,n)*2^n,n=0..19); # Zerinvary Lajos, Dec 08 2007

Table[2^n Binomial[2n,n],{n,0,30}] (* Harvey P. Dale, Dec 16 2014 *)

{a(n)=if(n<0, 0, 2^n*(2*n)!/n!^2)} /* Michael Somos, Jan 31 2007 */

{ for (n = 0, 200, write("b059304.txt", n, " ", 2^n * (2*n)! / n!^2); ) } \\ Harry J. Smith, Jun 25 2009

/* as lattice paths: same as in A092566 but use */
steps=[[1, 0], [1, 0], [0, 1]]; /* note the double [1, 0] */
/* Joerg Arndt, Jul 01 2011 */

def A059304(n): return pow(2,n)*binomial(2*n,n)
print([A059304(n) for n in range(41)]) # G. C. Greubel, Jan 18 2025

a(n) = 2^n * C(2*n,n).
D-finite with recurrence a(n) = 4*(2-1/n)*a(n-1).
a(n) = A000079(n)*A000984(n)
G.f.: 1/sqrt(1-8*x) - T. D. Noe, Jun 11 2002
E.g.f.: exp(4*x)*BesselI(0, 4*x). - Vladeta Jovovic, Aug 20 2003
a(n) = A038207(n,n). - Joerg Arndt, Jul 01 2011
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 4*x*(2*k+1)/(4*x*(2*k+1) + (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
E.g.f.: E(0)/2, where E(k) = 1 + 1/(1 - 4*x/(4*x + (k+1)^2/(2*k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
G.f.: Q(0)/(1+2*sqrt(x)), where Q(k) = 1 + 2*sqrt(x)/(1 - 2*sqrt(x)*(2*k+1)/(2*sqrt(x)*(2*k+1) + (k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 09 2013
O.g.f.: hypergeom([1/2], [], 8*x). - Peter Luschny, Oct 08 2015
a(n) = Sum_{k = 0..2*n} (-1)^(n+k)*binomial(2*n,k)*binomial(3*n-2*k,n)* binomial(n+k,n). - Peter Bala, Aug 04 2016
a(n) ~ 8^n/sqrt(Pi*n). - Ilya Gutkovskiy, Aug 04 2016
From Amiram Eldar, Jul 21 2020: (Start)
Sum_{n>=0} 1/a(n) = 8/7 + 8*sqrt(7)*arcsin(1/sqrt(8))/49.
Sum_{n>=0} (-1)^n/a(n) = (8/27)*(3 - arcsinh(1/sqrt(8))). (End)
a(n) = Sum_{k = n..2*n} binomial(2*n,k)*binomial(k,n). In general, for m >= 1, Sum_{k = n..m*n} binomial(m*n,k)*binomial(k,n) = 2^((m-1)*n)*binomial(m*n,n). - Peter Bala, Mar 25 2023
Conjecture: a(n) = Sum_{0 <= j, k <= n} binomial(n, j)*binomial(n, k)* binomial(k+j, n). - Peter Bala, Jul 16 2024

A034171 Related to triple factorial numbers A007559(n+1).

Original entry on oeis.org

1, 6, 42, 315, 2457, 19656, 160056, 1320462, 11003850, 92432340, 781473420, 6642524070, 56716936290, 486145168200, 4180848446520, 36059817851235, 311811366125385, 2702365173086670, 23467908082068450, 204170800313995515, 1779202688450532345, 15527587099204645920

Offset: 0

Wolfdieter Lang

Working with an offset of 1, we conjecture a(p*n) = a(n) (mod p^2) for prime p = 1 (mod 3) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/3 and 1 <= k <= (p-1)/3. Cf. A298799, A004981 and A004982. - Peter Bala, Dec 23 2019

Michael De Vlieger, Table of n, a(n) for n = 0..1050
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A007559, A025748, A034164, A035529.

Cf. A298799, A004981, A004982, A004987, A073005.

CoefficientList[Series[(-1 + (1 - 9 x)^(-1/3))/(3 x), {x, 0, 19}], x] (* Michael De Vlieger, Oct 13 2019 *)

a(n) = 3^n*A007559(n+1)/(n+1)! where A007559(n+1)=(3*n+1)!!!.
G.f.: (-1+(1-9*x)^(-1/3))/(3*x).
a(n) = A035529(n+1, 1) (first column of triangle).
Convolution of A004987(n) with A025748(n+1), n >= 0.
From R. J. Mathar, Jan 28 2020: (Start)
D-finite with recurrence: (n+1)*a(n) + 3*(-3*n-1)*a(n-1) = 0.
G.f.: (1F0(1/3;;9*x)-1)/(3*x). (End)
Sum_{n>=0} 1/a(n) = 3/8 + 3*sqrt(3)*Pi/32 + 9*log(3)/32. - Amiram Eldar, Dec 22 2022
a(n) ~ 3^(2*n+1) * n^(-2/3) / Gamma(1/3). - Amiram Eldar, Aug 19 2025

A004982 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 3).

Original entry on oeis.org

1, 6, 42, 308, 2310, 17556, 134596, 1038312, 8046918, 62587140, 488179692, 3816677592, 29897307804, 234578876616, 1843119744840, 14499208659408, 114181268192838, 900017055167076, 7100134546318044, 56053693786721400, 442824180915099060, 3500419715805068760, 27685137752276452920

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Conjecture: a(p*n) = a(n) (mod p^2) for prime p == 1 (mod 4) and all positive integers n. Cf. A004981. - Peter Bala, Dec 22 2019

Alois P. Heinz, Table of n, a(n) for n = 0..400

Main diagonal of A067001. Cf. A004981.

List([0..25], n-> 2^n*Product([0..n-1], k-> 4*k+3)/Factorial(n) ); # G. C. Greubel, Aug 22 2019

[1] cat [2^n*&*[4*k+3: k in [0..n-1]]/Factorial(n): n in [1..25]]; // G. C. Greubel, Aug 22 2019

A004982 := n -> (-8)^n*binomial(-3/4, n):
seq(A004982(n), n=0..25); # Peter Luschny, Oct 23 2018

Table[2^n/n! Product[4k+3,{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, Oct 03 2011 *)
Table[Sum[2^k*Binomial[2*n-2*k,n-k]*Binomial[n+k,n],{k,0,n}],{n,0,25}] (* Vaclav Kotesovec, Sep 15 2013 *)
FullSimplify[Table[8^n*Gamma[n+3/4]/(n!*Gamma[3/4]), {n, 0, 25}]] (* Vaclav Kotesovec, Sep 15 2013 *)
max = 30; s = Hypergeometric1F1[3/4, 1, 8x] + O[x]^(max+1);
CoefficientList[s, x]*(Range[0, max]!) (* Jean-François Alcover, Dec 19 2015, after Karol A. Penson *)

a(n)=2^n/n!*prod(k=0,n-1,4*k+3)
for(n=0,25,print(a(n)))

x='x+O('x^66); Vec((1-8*x)^(-3/4)) \\ Joerg Arndt, Apr 20 2013

[8^n*rising_factorial(3/4, n)/factorial(n) for n in (0..25)] # G. C. Greubel, Aug 22 2019

G.f.: (1 - 8*x)^(-3/4).
a(n) ~ Gamma(3/4)^-1*n^(-1/4)*2^(3*n)*{1 - 3/32*n^-1 + ...}
a(n) = 8^n*Gamma(n+3/4)/(n!*Gamma(3/4)). - Vaclav Kotesovec, Sep 15 2013
From Karol A. Penson, Dec 19 2015: (Start)
a(n) = (-8)^n*binomial(-3/4,n).
E.g.f.: is the hypergeometric function of type 1F1, in Maple notation hypergeom([3/4], [1], 8*x).
Representation as n-th moment of a positive function on (0, 8): a(n) = Integral_{x=0..8} ( x^n*2^(1/4)/(8*Pi*x^(1/4)*(1-x/8)^(3/4)) ) dx, n >= 0. This function is the solution of the Hausdorff moment problem on (0, 8) with moments equal to a(n). As a consequence this representation is unique. (End)
D-finite with recurrence: n*a(n) +2*(-4*n+1)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

More terms from Rick L. Shepherd, Mar 03 2002

A004984 a(n) = (2^n/n!)Product_{k=0..n-1} (4k - 1).

Original entry on oeis.org

1, -2, -6, -28, -154, -924, -5852, -38456, -259578, -1788204, -12517428, -88759944, -636112932, -4599585816, -33511268088, -245749299312, -1812401082426, -13433090375628, -100001895018564, -747382583822952, -5605369378672140, -42173731515723720

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A004981, A025749, A034256, A048779, A303538.

List([0..25],n->(2^n/Factorial(n))*Product([0..n-1],k->4*k-1)); # Muniru A Asiru, Apr 28 2018

[1] cat [2^n*(&*[4*k-1: k in [0..n-1]])/Factorial(n): n in [1..30]]; // G. C. Greubel, Aug 22 2019

seq(-(1/4)*8^n*GAMMA(n-1/4)/(GAMMA(3/4)*n!), n=0..30); # Robert Israel, Sep 29 2014

Table[8^n*Pochhammer[-1/4, n]/n!, {n,0,30}] (* G. C. Greubel, Aug 22 2019 *)
CoefficientList[Series[Surd[1-8x,4],{x,0,30}],x] (* Harvey P. Dale, Dec 08 2019 *)

a(n):=-1/n*(if n=1 then (2)^n else sum(sum(binomial(k,j)* binomial(j,n-1-3*k+2*j)*(3/2)^(3*k-n+1-j)*(-1)^(n-1-3*k)*(1/4)^(k-j),j,0,k)*binomial(k+n-1,n-1)*(2)^n,k,1,n-1)); /* Vladimir Kruchinin, Sep 14 2010 */

a(n):=binomial(1/4,n)*(-8)^n; /* Tani Akinari, Sep 28 2014 */

for(n=0,28,print1(2^n/n!*prod(k=0,n-1,(4*k-1)),","))

[8^n*rising_factorial(-1/4, n)/factorial(n) for n in (0..30)] # G. C. Greubel, Aug 22 2019

G.f.: (1 - 8*x)^(1/4).
a(n) ~ -1/4*Gamma(3/4)^-1*n^(-5/4)*2^(3*n)*{1 + 5/32*n^-1 + ...}
a(n) = -1/n*Sum_{k=1..n-1} 2^n*binomial(k+n-1, n-1) * ( Sum_{j=0..k} binomial(k,j)*binomial(j,n-1-3*k+2*j)*(3/2)^(3*k-n+1-j)*(-1)^(n-1-3*k)*(1/4)^(k-j) ), n>1. - Vladimir Kruchinin, Sep 14 2010
a(n) = 8^n*Pochhammer(-1/4, n)/n! = -(1/4)*8^n*Gamma(n-1/4)/(Gamma(3/4)*n!). - Robert Israel, Sep 29 2014
D-finite with recurrence: n*a(n) +2*(-4*n+5)*a(n-1)=0. - R. J. Mathar, Jan 16 2020

More terms from Jason Earls, Dec 04 2001

A214764 G.f. satisfies: A(x) = 1/A(-x*A(x)^4).

Original entry on oeis.org

1, 2, 10, 60, 390, 2660, 18772, 138984, 1107686, 9576100, 87944188, 830857464, 7876505340, 73967614584, 685644645896, 6289047266480, 57465415636166, 528315307772004, 4947263762389484, 47785581838822232, 480797992896880788, 5058812497153271912

Offset: 0

Paul D. Hanna, Jul 29 2012

nonn

Compare to: W(x) = 1/W(-x*W(x)^4) when W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.
Compare to: B(x) = 1/B(-x*B(x)^4) when B(x) = 1/(1-8*x)^(1/4) = g.f. of A004981.
An infinite number of functions G(x) satisfy (*) G(x) = 1/G(-x*G(x)^4); for example, (*) is satisfied by G(x) = W(m*x), where W(x) = Sum_{n>=0} (2*n+1)^(n-1)*x^n/n!.

			G.f.: A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 390*x^4 + 2660*x^5 + 18772*x^6 +...
A(x)^4 = 1 + 8*x + 64*x^2 + 512*x^3 + 4096*x^4 + 32800*x^5 + 263168*x^6 +...

Cf. A214761, A214762, A214763, A214765, A214766, A214767, A214768, A214769.

{a(n)=local(A=1+2*x);for(i=0,n,A=(A+1/subst(A,x,-x*A^4+x*O(x^n)))/2);polcoeff(A,n)}
for(n=0,31,print1(a(n),", "))

The g.f. of this sequence is the limit of the recurrence:

(*) G_{n+1}(x) = (G_n(x) + 1/G_n(-x*G_n(x)^4))/2 starting at G_0(x) = 1+2*x.

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

A127776 a(n) = ( (2^n / n!) * Product_{k=0..n-1} (4*k + 1) )^2.

Original entry on oeis.org

1, 4, 100, 3600, 152100, 7033104, 344622096, 17582760000, 924193822500, 49701090010000, 2721631688947600, 151241747739534400, 8507348310348810000, 483459012855561960000, 27715027900230072360000, 1600820011517288979513600, 93072675982122379574532900

Offset: 0

Michael Somos, Jan 14 2007

nonn

Cayley (1878) refers to Gauss, Werke, t. iii, p. 424 for a slightly different form of a square of a hypergeometric series being hypergeometric. - Michael Somos, Jun 25 2012

			G.f. = 1 + 4*x + 100*x^2 + 3600*x^3 + 152100*x^4 + 7033104*x^5 + ...

A. Cayley, An Identity, Messenger of Mathematics, 7 (1878), p. 69

Gheorghe Coserea, Table of n, a(n) for n = 0..301
Robert S. Maier, On Rationally Parametrized Modular Equations, arXiv:math/0611041 [math.NT], 2006.

Cf. A002897, A004981, A091401, A186284.

[1] cat [n le 1 select 4 else (4*(4*n-3)^2*Self(n-1)/ n^2): n in [1..30]]; // Vincenzo Librandi, Aug 26 2016

From Michael Somos, Jun 25 2012 (Start)
a[ n_] := If[ n < 0, 0, (Pochhammer[ 1/4, n] 8^n / n!)^2];
a[ n_] := SeriesCoefficient[ Hypergeometric2F1[ 1/4, 1/4, 1, 64 x], {x, 0, n}];
a[ n_] := SeriesCoefficient[ HypergeometricPFQ[ {1/2, 1/2, 1/2}, {1, 1}, 64 x]^(1/2), {x, 0, n}];
(End)
Join[{1}, RecurrenceTable[{a[1] == 4, a[n] == (4 (4 n - 3)^2 a[n-1] / n^2)}, a, {n, 20}]] (* Vincenzo Librandi, Aug 26 2016 *)

{a(n) = if( n<0, 0, prod( k=1, n, (8*k - 6) / k)^2)};

{a(n) = my(A); if( n<1, n==0, A = x * O(x^n); polcoeff( subst( 1 / agm(1, sqrt(1 - 16*x + A) ), x, serreverse( x*(1 - 16*x) + A )), n))};

seq(N) = {
  my(a = vector(N)); a[1] = 4;
  for (n=2, N, a[n] = 4*(4*n-3)^2*a[n-1]/n^2);
  concat(1,a);
};
seq(15) \\ Gheorghe Coserea, Aug 26 2016

Expansion of K(k) / (Pi/2) in powers of (k * k'/4)^2, where K(k) is the complete elliptic integral of first kind evaluated at modulus k.
Expansion of 1 / AGM( 1, (1 - 16*x)^(1/2) ) in powers of x * (1 - 16*x) where AGM() is the arithmetic-geometric mean.
G.f.: F(1/4, 1/4; 1; 64*x).
a(n) = A004981(n)^2. Convolution square is A002897.
a(n) ~ 64^n / (Gamma(1/4)^2 * n^(3/2)). - Vaclav Kotesovec, Sep 08 2015
From Gheorghe Coserea, Aug 26 2016: (Start)
n^2 * a(n) = 4*(4*n-3)^2 * a(n-1), with a(0) = 1.
0 = 16*x*(x+64)*y'' + 8*(3*x+128)*y' + y, where y(x) = A(x/-4096). (End)

A224881 Expansion of 1/(1 - 16*x)^(1/8).

Original entry on oeis.org

1, 2, 18, 204, 2550, 33660, 460020, 6440280, 91773990, 1325624300, 19354114780, 285033326760, 4227994346940, 63094684869720, 946420273045800, 14259398780556720, 215673406555920390, 3273161111260438860, 49824785804742235980, 760483572809223601800

Offset: 0

Paul D. Hanna, Jul 23 2013

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 204*x^3 + 2550*x^4 + 33660*x^5 + ...
where
A(x)^8 = 1 + 16*x + 256*x^2 + 4096*x^3 + 65536*x^4 + ... + 16^n*x^n + ...
Also,
A(x)^4 = 1 + 8*x + 96*x^2 + 1280*x^3 + 17920*x^4 + 258048*x^5 + ... + 4^n*A000984(n)*x^n + ...
A(x)^2 = 1 + 4*x + 40*x^2 + 480*x^3 + 6240*x^4 + 84864*x^5 + ... + 2^n*A004981(n)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..500

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), this sequence (b=16), A034688 (b=25), A298799 (b=27), A004993 (b=36), A034835 (b=49).

Cf. A301271.

List([0..20],n->(2^n/Factorial(n))*Product([0..n-1],k->8*k+1)); # Muniru A Asiru, Jun 23 2018

seq(coeff(series(1/(1-16*x)^(1/8), x,50),x,n+1),n=0..20); # Muniru A Asiru, Jun 23 2018

CoefficientList[Series[1/(1-16*x)^(1/8), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 24 2013 *)

{a(n)=polcoeff(1/(1-16*x +x*O(x^n))^(1/8),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=(2^n/n!)*prod(k=0,n-1,8*k + 1)}
for(n=0,30,print1(a(n),", "))

a(n) = (2^n/n!) * Product_{k=0..n-1} (8*k + 1).

a(n) ~ 16^n/(GAMMA(1/8)*n^(7/8)). - Vaclav Kotesovec, Jul 24 2013

cp's OEIS Frontend

A208890 a(n) = A000984(n)*A004981(n), the term-wise product of the coefficients in (1-4*x)^(-1/2) and (1-8*x)^(-1/4).

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A007696 Quartic (or 4-fold) factorial numbers: a(n) = Product_{k = 0..n-1} (4*k + 1).

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A059304 a(n) = 2^n * (2*n)! / (n!)^2.

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A034171 Related to triple factorial numbers A007559(n+1).

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A004982 a(n) = (2^n/n!) * Product_{k=0..n-1} (4*k + 3).

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A004984 a(n) = (2^n/n!)*Product_{k=0..n-1} (4*k - 1).

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A208890 a(n) = A000984(n)A004981(n), the term-wise product of the coefficients in (1-4x)^(-1/2) and (1-8*x)^(-1/4).

A004984 a(n) = (2^n/n!)Product_{k=0..n-1} (4k - 1).