A034977 -id:A034977 - cp's OEIS frontend

A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

1, 4, 40, 480, 6240, 84864, 1188096, 16972800, 246105600, 3609548800, 53421322240, 796463349760, 11946950246400, 180123249868800, 2727580640870400, 41459225741230080, 632253192553758720

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..500
A. Straub, V. H. Moll, and T. Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009) 31-41, eq (1.10).

Cf. A007696.

Expansion of (1-b^2*x)^(-1/b): A000984 (b=2), A004987 (b=3), this sequence (b=4), A034688 (b=5), A004993 (b=6), A034835 (b=7), A034977 (b=8), A035024 (b=9), A035308 (b=10).

CoefficientList[Series[1/Surd[1-16x,4],{x,0,20}],x] (* Harvey P. Dale, Aug 06 2018 *)

a(n) = (4^n/n!)*A007696(n), n >= 1, a(0) := 1, A007696(n) = (4*n-3)!^4 := Product_{j = 1..n} 4*j - 3.
G.f.: (1 - 16*x)^(-1/4).
D-finite with recurrence: n*a(n) + 4*(-4*n + 3)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-16)^n*binomial(-1/4, n).
a(n) ~ Gamma(3/4)/(sqrt(2)*Pi) * 16^n/n^(3/4).
E.g.f.: hypergeom([1/4], [1], 16*x).
a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-1/4, k)* binomial(-1/4, 2*n - k).
(16^n)*a(n) = Sum_{k = 0..2*n} (-1)^k*a(k)*a(2*n-k).
Sum_{k = 0..n} a(k)*a(n-k) = (4^n)*binomial(2*n, n) = A098430.
Sum_{k = 0..2*n} a(k)*a(2*n-k) = (16^n)*binomial(4*n, 2*n). (End)

A386271 Expansion of 1/(1 - 49*x)^(2/7).

Original entry on oeis.org

1, 14, 441, 16464, 662676, 27832392, 1201431588, 52862989872, 2359010923038, 106417603861492, 4842000975697886, 221851681068339504, 10223664969232645476, 473434331652157890504, 22014696421825341908436, 1027352499685182622393680, 48092938891512611510804145

Offset: 0

Seiichi Manyama, Jul 17 2025

Cf. A020918 (k=2, m=7), A020920 (k=2, m=9), A034835 (k=7, m=1), A034977 (k=8, m=1), A035024 (k=9, m=1), A216702 (k=4, m=3), A216703 (k=7, m=6), A354019 (k=6, m=1), this sequence (k=7, m=2), A386272 (k=7, m=3), A386273 (k=7, m=4), A386274 (k=7, m=5).

my(N=20, x='x+O('x^N)); Vec(1/(1-49*x)^(2/7))

a(n) = (-49)^n * binomial(-2/7,n).
a(n) = 7^n/n! * Product_{k=0..n-1} (7*k+2).
a(n) = 7^n * Product_{k=1..n} (7 - 5/k).
In general, 1/(1 - k^2*x)^(m/k) leads to the D-finite recurrence k*(k*n-k+m)*a(n-1) - n*a(n) = 0. This sequence is case k=7, m=2: (49*n-35)*a(n-1) - n*a(n) = 0. - Georg Fischer, Jul 19 2025

A248332 Square array read by antidiagonals downwards: super Patalan numbers of order 8.

Original entry on oeis.org

1, 8, 56, 288, 224, 3360, 13056, 5376, 8960, 206080, 652800, 182784, 161280, 412160, 12776960, 34467840, 7311360, 4386816, 5935104, 20443136, 797282304, 1884241920, 321699840, 146227200, 134529024, 245317632, 1063043072, 49963024384, 105517547520, 15073935360, 5514854400, 3843686400

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers, A068555, based on Patalan numbers of order 8, A025753.

			T(0..4,0..4) is
  1           8           288         13056       652800
  56          224         5376        182784      7311360
  3360        8960        161280      4386816     146227200
  206080      412160      5935104     134529024   3843686400
  12776960    20443136    245317632   4766171136  119154278400

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, A025753, A034977 (first row), A216704 (first column), A248324, A248325, A248326, A248328, A248329.

matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*64^(n+k)*binomial(n-1/8,n+k)) \\ Michel Marcus, Oct 09 2014

T(0,0)=1, T(n,k) = T(n-1,k)*(64*n-8)/(n+k), T(n,k) = T(n,k-1)*(64*k-56)/(n+k).
G.f.: (x/(1-64*x)^(7/8)+y/(1-64*y)^(1/8))/(x+y-64*x*y).
T(n,k) = (-1)^k*64^(n+k)*binomial(n-1/8,n+k).

A034996 Related to octo-factorial numbers A045755.

Original entry on oeis.org

1, 36, 1632, 81600, 4308480, 235530240, 13189693440, 751812526080, 43438057062400, 2536782532444160, 149439552820346880, 8866746800673914880, 529276578255612149760, 31756594695336728985600, 1913864106972293533532160, 115788778471823758778695680, 7029059963701301121153761280

Offset: 1

Wolfdieter Lang

Convolution of A034977(n-1) with A025753(n), n >= 1.

Michael De Vlieger, Table of n, a(n) for n = 1..555
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.

Cf. A045755, A034977, A025753, A034904, A203142.

Rest@ CoefficientList[Series[(-1 + (1 - 64*x)^(-1/8))/8, {x, 0, 14}], x] (* Michael De Vlieger, Oct 13 2019 *)

a(n) = 8^(n-1)*A045755(n)/n!, where A045755(n) = (8*n-7)!^8 = Product_{j=1..n} (8*j-7).
G.f.: (-1+(1-64*x)^(-1/8))/8.
D-finite with recurrence: n*a(n) + 8*(-8*n+7)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 8^(2*n-1) * n^(-7/8) / Gamma(1/8). - Amiram Eldar, Aug 18 2025

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A034385 Expansion of (1-16*x)^(-1/4), related to quartic factorial numbers.

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A386271 Expansion of 1/(1 - 49*x)^(2/7).

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A248332 Square array read by antidiagonals downwards: super Patalan numbers of order 8.

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A034996 Related to octo-factorial numbers A045755.

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