A048779 -id:A048779 - cp's OEIS frontend

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

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

A025749 4th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 6, 56, 616, 7392, 93632, 1230592, 16612992, 228890112, 3204461568, 45445091328, 651379642368, 9419951751168, 137262154088448, 2013178259963904, 29694379334467584, 440175505428578304, 6553724191936610304, 97960930026841964544, 1469413950402629468160

Offset: 0

Olivier Gérard

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seq., Vol. 3 (2000), Article 00.2.4.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Thomas M. Richardson, The Super Patalan Numbers, J. Int. Seq. 18 (2015), Article 15.3.3; arXiv preprint, arXiv:1410.5880 [math.CO], 2014.

Cf. A034176, A048779, A068465.

a[n_] := (4^(n-1)*Sum[ Binomial[n+k-1, n-1]*Sum[ Binomial[j, n-3*k+2*j-1] * 4^(j-k) * Binomial[k, j] * 3^(-n+3*k-j+1) * 2^(n-3*k+j-1) * (-1)^(n-3*k+2*j-1), {j, 0, k}], {k, 1, n-1}])/n; a[0] = a[1] = 1; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Mar 05 2013, after Vladimir Kruchinin *)
a[n_] := 16^(n-1) * Pochhammer[3/4, n-1]/n!; a[0] = 1; Array[a, 21, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n):=(4^(n-1)*sum(binomial(n+k-1,n-1)*sum(binomial(j,n-3*k+2*j-1)*4^(j-k)*binomial(k,j)*3^(-n+3*k-j+1)*2^(n-3*k+j-1)*(-1)^(n-3*k+2*j-1),j,0,k),k,1,n-1))/n; /* Vladimir Kruchinin, Apr 01 2011 */

a(n) = 2^(n-1) * A048779(n), n > 1.
From Wolfdieter Lang: (Start)
G.f.: (5-(1-16*x)^(1/4))/4.
a(n) = 4^(n-1)*3*A034176(n-1)/n!, n >= 2, where 3*A034176(n-1) = (4*n-5)(!^4) = Product_{j=2..n} (4*j - 5). (End)
a(n) = (4^(n-1) * Sum_{k=1..n-1} binomial(n+k-1,n-1) * Sum_{j=0..k} binomial(j,n-3*k+2*j-1)*4^(j-k)*binomial(k,j)*3^(-n+3*k-j+1)*2^(n-3*k+j-1)*(-1)^(n-3*k+2*j-1))/n. - Vladimir Kruchinin, Apr 01 2011
n*a(n) + 4*(-4*n+5)*a(n-1) = 0. - R. J. Mathar, Apr 05 2018
a(n) ~ 16^(n-1) / (Gamma(3/4) * n^(5/4)). - Amiram Eldar, Aug 20 2025

A254286 Expansion of (1 - (1-256x)^(1/4)) / (64x).

Original entry on oeis.org

1, 96, 14336, 2523136, 484442112, 98180268032, 20645907791872, 4459516083044352, 983075545417777152, 220208922173582082048, 49967406340478261526528, 11459191854083014643417088, 2651480699775516003646046208, 618173786004806016850049630208

Offset: 0

Vaclav Kotesovec, Jan 27 2015

G. C. Greubel, Table of n, a(n) for n = 0..250

Cf. A000108, A008545, A025749, A048779, A254282, A254287.

[Round(2^(8*n)*Gamma(n+3/4)/(Gamma(3/4)*Gamma(n+2))): n in [0..30]]; // G. C. Greubel, Aug 10 2022

CoefficientList[Series[(1-(1-256*x)^(1/4)) / (64*x),{x,0,20}],x]
CoefficientList[Series[Hypergeometric1F1[3/4,2,256*x],{x,0,20}],x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2015 *)

[2^(8*n)*rising_factorial(3/4,n)/factorial(n+1) for n in (0..30)] # G. C. Greubel, Aug 10 2022

G.f.: (1 - (1-256*x)^(1/4)) / (64*x).
a(n) ~ 256^n / (Gamma(3/4) * n^(5/4)).
Recurrence: (n+1)*a(n) = 64*(4*n-1)*a(n-1).
a(n) = 256^n * Gamma(n+3/4) / (Gamma(3/4) * Gamma(n+2)).
E.g.f.: hypergeom([3/4], [2], 256*x). - Vaclav Kotesovec, Jan 28 2015
From Peter Bala, Sep 01 2017: (Start)
a(n) = (-1)^n*binomial(1/4, n+1)*4^(4*n+1). Cf. A000108(n) = (-1)^n*binomial(1/2, n+1)*2^(2*n+1).
a(n) = 16^n*A025749(n+1); a(n) = 32^n*A048779(n+1).
(End)

A097179 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 4^n, where R_n(y) forms the initial (n+1) terms of g.f. A077860(y)^(n+1).

Original entry on oeis.org

1, 1, 6, 1, 9, 42, 1, 12, 74, 308, 1, 15, 115, 595, 2310, 1, 18, 165, 1020, 4746, 17556, 1, 21, 224, 1610, 8722, 37730, 134596, 1, 24, 292, 2392, 14778, 73080, 299508, 1038312, 1, 27, 369, 3393, 23535, 130851, 604707, 2376099, 8046918

Offset: 0

Paul D. Hanna, Aug 02 2004

Row sums form A097180. Diagonal is A004982. Ratio of g.f.s of any two adjacent diagonals equals g.f. of A048779, where the g.f.s satisfy: A077860(x*A048779(x)) = A048779(x).

			Row polynomials evaluated at y=1/2 equals powers of 4:
4^1 = 1 + 6/2;
4^2 = 1 + 9/2 + 42/2^2;
4^3 = 1 + 12/2 + 74/2^2 + 308/2^3;
4^4 = 1 + 15/2 + 115/2^2 + 595/2^3 + 2310/2^4;
where A077860(y)^(n+1) has the same initial terms as the n-th row:
A077860(y) = 1 +3*y +5*y^2 +5*y^3 +1*y^4 -7*y^5 -15*y^6 -15*y^7 +...
A077860(y)^2 = 1 + 6*y +...
A077860(y)^3 = 1 + 9*y + 42*y^2 +...
A077860(y)^4 = 1 + 12*y + 74*y^2 + 308*y^3 +...
A077860(y)^5 = 1 + 15*y + 115*y^2 + 595*y^3 + 2310*y^4 +...
Rows begin with n=0:
  1;
  1,  6;
  1,  9,  42;
  1, 12,  74,  308;
  1, 15, 115,  595,  2310;
  1, 18, 165, 1020,  4746, 17556;
  1, 21, 224, 1610,  8722, 37730,  134596;
  1, 24, 292, 2392, 14778, 73080,  299508, 1038312;
  1, 27, 369, 3393, 23535, 130851, 604707, 2376099, 8046918; ...

G. C. Greubel, Rows n = 0..50 of triangle, flattened

Cf. A004982, A048779, A077860, A097180, A097181.

Table[SeriesCoefficient[2*y/((1-8*x*y) +(2*y-1)*(1-8*x*y)^(3/4)), {x, 0, n}, {y,0,k}], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Sep 17 2019 *)

{T(n,k)=if(n==0,1,if(k==0,1,if(k==n, 2^n*(4^n-sum(j=0,n-1, T(n,j)/2^j)), polcoeff((Ser(vector(n,i,T(n-1,i-1)), x) +x*O(x^k))^((n+1)/n),k,x))))}

G.f.: A(x, y) = 2*y/((1-8*x*y) + (2*y-1)*(1-8*x*y)^(3/4)).

G.f.: A(x, y) = A004982(x*y)/(1 - x*A048779(x*y)).

cp's OEIS Frontend

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

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A025749 4th-order Patalan numbers (generalization of Catalan numbers).

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Maxima

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A254286 Expansion of (1 - (1-256*x)^(1/4)) / (64*x).

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A097179 Triangle read by rows in which row n gives coefficients of polynomial R_n(y) that satisfies R_n(1/2) = 4^n, where R_n(y) forms the initial (n+1) terms of g.f. A077860(y)^(n+1).

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

A254286 Expansion of (1 - (1-256x)^(1/4)) / (64x).