A025749 -id:A025749 - cp's OEIS frontend

A049213 A convolution triangle of numbers obtained from A025749.

1, 6, 1, 56, 12, 1, 616, 148, 18, 1, 7392, 1904, 276, 24, 1, 93632, 25312, 4080, 440, 30, 1, 1230592, 344960, 59808, 7360, 640, 36, 1, 16612992, 4792128, 876960, 118224, 11960, 876, 42, 1, 228890112, 67586816, 12900416, 1860992, 209200, 18096, 1148

Offset: 1

Wolfdieter Lang

a(n,1) = A025749(n); a(n,1)= 4^(n-1)*3*A034176(n-1)/n!, n >= 2.

G.f. for m-th column: ((1-(1-16*x)^(1/4))/4)^m.

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

Cf. A048966. Row sums = A025757.

a[n_, n_] = 1; a[n_, m_] := m/n * 4^(n-m) * Sum[ Binomial[n+k-1, n-1] * Sum[ Binomial[j, n-m-3*k+2*j] * 4^(j-k) * Binomial[k, j] * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j), {j, 0, k}], {k, 1, n-m}]; Table[a[n, m], {n, 1, 9}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jul 05 2013, after Vladimir Kruchinin *)

a(n, m) = 4*(4*(n-1)-m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n < m; a(n, 0) := 0; a(1, 1)=1.

a(n,m) = (m/n) * 4^(n-m) * Sum_{k=1..n-m} binomial(n+k-1, n-1) * Sum_{j=0..k} binomial(j, n-m-3*k+2*j) * 4^(j-k) * binomial(k,j) * 3^(-n+m+3*k-j) * 2^(n-m-3*k+j) * (-1)^(n-m-3*k+2*j), n > m; a(n,n)=1. - Vladimir Kruchinin, Feb 08 2011

A034176 One third of quartic factorial numbers.

Original entry on oeis.org

1, 7, 77, 1155, 21945, 504735, 13627845, 422463195, 14786211825, 576662261175, 24796477230525, 1165434429834675, 59437155921568425, 3269043575686263375, 192873570965489539125, 12151034970825840964875, 814119343045331344646625, 57802473356218525469910375

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..350
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), Article 10.6.7, p 39.

Cf. A007696, A034177, A034256, A025749.

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

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

A034176:=n->`if`(n=1, 1, (4*n-1)*A034176(n-1)); seq(A034176(n), n=1..20); # G. C. Greubel, Aug 15 2019

Table[4^n*Pochhammer[3/4, n]/3, {n, 20}] (* G. C. Greubel, Aug 15 2019 *)

m=20; v=concat([1], vector(m-1)); for(n=2, m, v[n]=(4*n-1)*v[n-1]); v \\ G. C. Greubel, Aug 15 2019

[4^n*rising_factorial(3/4, n)/3 for n in (1..20)] # G. C. Greubel, Aug 15 2019

3*a(n) = (4*n-1)(!^4) := Product_{j=1..n} 4*j-1 = (4*n-1)!!/A007696(n) = (4*n)!/(4^n*(2*n)!*A007696(n)), A007696(n)=(4*n-3)(!^4), n >= 1;
E.g.f.: (-1 + (1-4*x)^(-3/4))/3.
a(n) ~ 4/3 * 2^(1/2) * Pi^(1/2) * Gamma(3/4)^(-1) * n^(5/4) * 2^(2*n) * e^(-n) * n^n * {1 + 71/96*n^(-1) + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
G.f.: 1/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
D-finite with recurrence: a(n) + (-4*n+1) * a(n-1) = 0. - R. J. Mathar, Feb 24 2020
Sum_{n>=1} 1/a(n) = 3*exp(1/4)*(Gamma(3/4) - Gamma(3/4, 1/4)) / sqrt(2). - Amiram Eldar, Dec 18 2022
a(n) = 4^(n-1) * Gamma(n + 3/4) / Gamma(7/4). - Peter McNair, May 06 2024

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

A034255 Related to quartic factorial numbers A007696.

Original entry on oeis.org

1, 10, 120, 1560, 21216, 297024, 4243200, 61526400, 902387200, 13355330560, 199115837440, 2986737561600, 45030812467200, 681895160217600, 10364806435307520, 158063298138439680, 2417438677411430400, 37067393053641932800, 569667303771760230400, 8772876478085107548160

Offset: 1

Wolfdieter Lang

Michael De Vlieger, Table of n, a(n) for n = 1..833
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.
Index entries for sequences related to factorial numbers.

First column of triangle A048882.

Cf. A007696, A068466.

Rest[CoefficientList[Series[(-1+(1-16x)^(-1/4))/4,{x,0,20}],x]] (* Harvey P. Dale, May 19 2011 *)

a(n) = 4^(n-1)*A007696(n)/n!, where A007696(n) = (4*n-3)(!^4) = Product_{j=1..n} (4*j-3), n >= 1.
G.f.: (-1+(1-16*x)^(-1/4))/4.
a(n) = A048882(n, 1).
Convolution of A034385(n-1) with A025749(n), n >= 1.
D-finite with recurrence: n*a(n) + 4*(-4*n+3)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 2^(4*n-2) * n^(-3/4) / Gamma(1/4). - Amiram Eldar, Aug 18 2025

A248325 Square array read by antidiagonals downwards: super Patalan numbers of order 4.

Original entry on oeis.org

1, 4, 12, 40, 24, 168, 480, 160, 224, 2464, 6240, 1440, 1120, 2464, 36960, 84864, 14976, 8064, 9856, 29568, 561792, 1188096, 169728, 69888, 59136, 98560, 374528, 8614144, 16972800, 2036736, 678912, 439296, 506880, 1070080, 4922368, 132903936, 246105600, 25459200, 7128576, 3734016, 3294720, 4815360

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers of Gessel, A068555, based on Patalan numbers of order 4, A025749.

			T(0..4, 0..4) is:
  1      4      40     480    6240
  12     24     160    1440   14976
  168    224    1120   8064   69888
  2464   2464   9856   59136  439296
  36960  29568  98560  506880 3294720

Thomas M. Richardson, The Super Patalan Numbers, arXiv:1410.5880 [math.CO], 2014.
Thomas M. Richardson, The Super Patalan Numbers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.3.3.

Cf. A068555, A025749, A216702 (first column), A034385 (first row), A248324.

T(0,0)=1, T(n,k) = T(n-1,k)*(16*n-4)/(n+k), T(n,k) = T(n,k-1)*(16*k-12)/(n+k).

G.f.: (x/(1-16*x)^(3/4)+y/(1-16*y)^(1/4))/(x+y-16*x*y).

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)

A034256 Expansion of 2 - (1 - 16*x)^(1/4), related to quartic factorial numbers A034176.

Original entry on oeis.org

1, 4, 24, 224, 2464, 29568, 374528, 4922368, 66451968, 915560448, 12817846272, 181780365312, 2605518569472, 37679807004672, 549048616353792, 8052713039855616, 118777517337870336, 1760702021714313216, 26214896767746441216, 391843720107367858176, 5877655801610517872640

Offset: 0

Wolfdieter Lang

Cf. A025749, A034176, A068465.

a[n_] := 4^(2*n-1)*Pochhammer[3/4, n-1]/n!; a[0] = 1; Array[a, 25, 0] (* Amiram Eldar, Aug 19 2025 *)

Equals 4 * A025749(n), n > 0.
a(n) = 4^n*3*A034176(n-1)/n!, n >= 2, where 3*A034176(n-1) = (4*n-5)(!^4) = Product_{j=2..n} (4*j - 5).
O.g.f.: A(x) = 2 - (1 - 16*x)^(1/4).
From Peter Bala, Nov 19 2015: (Start)
For n >= 1, a(n) = (1/(sqrt(2)*Pi)) * Integral_{x = 0..16} x^(n-1)*((16 - x)/x)^(1/4) dx.
It appears that sqrt(A(x)) = 1 + 2*x + 10*x^2 + 92*x^3 + 998*x^4 + 11868*x^5 + 149316*x^6 + ... has integer coefficients. (End)
a(n) ~ 4^(2*n-1) * n^(-5/4) / Gamma(3/4). - Amiram Eldar, Aug 19 2025

More terms from Amiram Eldar, Aug 19 2025

cp's OEIS Frontend

A049213 A convolution triangle of numbers obtained from A025749.

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A034176 One third of quartic factorial numbers.

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

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A034255 Related to quartic factorial numbers A007696.

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A248325 Square array read by antidiagonals downwards: super Patalan numbers of order 4.

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

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A034256 Expansion of 2 - (1 - 16*x)^(1/4), related to quartic factorial numbers A034176.

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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).