A034385 -id:A034385 - cp's OEIS frontend

A034688 Expansion of (1-25*x)^(-1/5), related to quintic factorial numbers A008548.

1, 5, 75, 1375, 27500, 577500, 12512500, 277062500, 6233906250, 141994531250, 3265874218750, 75708902343750, 1766541054687500, 41445770898437500, 976936028320312500, 23120819336914062500, 549119459251708984375

Offset: 0

Wolfdieter Lang

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

Cf. A008548, A034385, A034687, A049380, A049381, A049382.

List([0..20], n-> 5^n*Product([0..n-1], k-> 5*k+1)/Factorial(n)); # G. C. Greubel, Aug 17 2019

[1] cat [5^n*(&*[5*k+1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 17 2019

A034688 := n -> (-25)^n*binomial(-1/5, n):
seq(A034688(n), n=0..16); # Peter Luschny, Oct 23 2018

Table[(-25)^n*Binomial[-1/5,n], {n,0,20}] (* G. C. Greubel, Aug 17 2019 *)
CoefficientList[Series[1/Surd[1-25x,5],{x,0,20}],x] (* Harvey P. Dale, Sep 11 2022 *)

vector(20, n, n--; 5^n*prod(k=0, n-1, 5*k+1)/n!) \\ G. C. Greubel, Aug 17 2019

[5^n*product(5*k+1 for k in (0..n-1))/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 17 2019

a(n) = (5^n/n!)*A008548(n), n >= 1, a(0) := 1, where A008548(n)=(5*n-4)(!^5) := Product_{j=1..n} (5*j-4).
G.f.: (1-25*x)^(-1/5).
a(n) ~ Gamma(1/5)^-1*n^(-4/5)*5^(2*n)*{1 - 2/25*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = (-25)^n*binomial(-1/5, n). - Peter Luschny, Oct 23 2018
E.g.f.: L_{-1/5}(25*x), where L_{k}(x) is the Laguerre polynomial. - Stefano Spezia, Aug 17 2019
D-finite with recurrence: n*a(n) +5*(-5*n+4)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

A216702 a(n) = Product_{k=1..n} (16 - 4/k).

Original entry on oeis.org

1, 12, 168, 2464, 36960, 561792, 8614144, 132903936, 2060011008, 32044615680, 499896004608, 7816555708416, 122459372765184, 1921670157238272, 30197673899458560, 475110069351481344, 7482983592285831168, 117967035454858985472, 1861257670509997326336

Offset: 0

Michel Lagneau, Sep 16 2012

This sequence is generalizable: Product_{k=1..n} (q^2 - q/k) = (q^n/n!) * Product_{k=0..n-1} (q*k + q-1) = expansion of (1- x*q^2)^((1-q)/q).

Harvey P. Dale, Table of n, a(n) for n = 0..800

Cf. A004988, A049382, A004994, A034385, A098430.

seq(product(16-4/k, k=1.. n), n=0..20);
seq((4^n/n!)*product(4*k+3, k=0.. n-1), n=0..20);

Table[Product[16-4/k,{k,n}],{n,0,20}] (* or *) CoefficientList[ Series[ 1/(1-16*x)^(3/4),{x,0,20}],x] (* Harvey P. Dale, Sep 19 2012 *)

G.f.: 1/(1-16*x)^(3/4). - Harvey P. Dale, Sep 19 2012
From Peter Bala, Sep 24 2023: (Start)
a(n) = 16^n * binomial(n - 1/4, n).
P-recursive: a(n) = 4*(4*n - 1)/n * a(n-1) with a(0) = 1. (End)
From Peter Bala, Mar 31 2024: (Start)
a(n) = (-16)^n * binomial(-3/4, n).
a(n) ~ 1/Gamma(3/4) * 16^n/n^(1/4).
E.g.f.: hypergeom([3/4], [1], 16*x).
a(n) = (16^n)*Sum_{k = 0..2*n} (-1)^k*binomial(-3/4, k)* binomial(-3/4, 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) = (16^n)/(2*n)! * Product_{k = 1..n} (4*k^2 - 1) =  (16^n)/(2*n)! * A079484(n). (End)

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

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).

A229558 E.g.f.: exp(x) / (2 - exp(4*x))^(1/4).

Original entry on oeis.org

1, 2, 12, 152, 2832, 69152, 2089152, 75204992, 3142025472, 149428961792, 7969790856192, 471098477484032, 30567292903821312, 2159857294035525632, 165083372031671058432, 13570774387950150582272, 1193933787763434969956352, 111932230270819401046556672

Offset: 0

Paul D. Hanna, Dec 18 2013

			E.g.f.: A(x) = 1 + 2*x + 12*x^2/2! + 152*x^3/3! + 2832*x^4/4! + 69152*x^5/5! +...
where A(x)^5 = 1 + 10*x + 140*x^2/2! + 2680*x^3/3! + 66320*x^4/4! +...
Also, A(x)^4 = 1 + 8*x + 96*x^2/2! + 1664*x^3/3! + 38400*x^4/4! +...
and log(A(x)) = 2*x + 8*x^2/2! + 96*x^3/3! + 1664*x^4/4! + 38400*x^5/5! +...

G. C. Greubel, Table of n, a(n) for n = 0..345

Cf. A124214, A034385, A124212.

CoefficientList[Series[E^x/(2-E^(4*x))^(1/4), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Dec 19 2013 *)

{a(n)=local(A=1+x,X=x+x*O(x^n));n!*polcoeff(exp(X)/(2-exp(4*X))^(1/4),n)}
for(n=0, 30, print1(a(n), ", "))

{a(n)=local(A=1+x); for(i=1, n, A=1+intformal(A+A^5+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

E.g.f. A(x) satisfies: A'(x) = A(x) + A(x)^5.
E.g.f. A(x) satisfies: A(x) = exp(x + Integral A(x)^4 dx).
a(n) ~ GAMMA(3/4) * 4^n * n^(n-1/4) / (sqrt(Pi) * exp(n) * log(2)^(n+1/4)). - Vaclav Kotesovec, Dec 19 2013
a(n) = 1/2^(1/4) * Sum_{k >= 0} (1/32)^k*A034385(k)*(4*k + 1)^n = 1/2^(1/4)*Sum_{k >= 0} (-1/2)^k*binomial(-1/4, k)*(4*k + 1)^n. Cf. A124212 and A124214. - Peter Bala, Aug 30 2016

cp's OEIS Frontend

A034688 Expansion of (1-25*x)^(-1/5), related to quintic factorial numbers A008548.

Views

Author

Keywords

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A216702 a(n) = Product_{k=1..n} (16 - 4/k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A034255 Related to quartic factorial numbers A007696.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A229558 E.g.f.: exp(x) / (2 - exp(4*x))^(1/4).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula