A034687 -id:A034687 - cp's OEIS frontend

A049375 A convolution triangle of numbers obtained from A034687.

1, 15, 1, 275, 30, 1, 5500, 775, 45, 1, 115500, 19250, 1500, 60, 1, 2502500, 471625, 44625, 2450, 75, 1, 55412500, 11495000, 1254000, 85000, 3625, 90, 1, 1246781250, 279675000, 34093125, 2698875, 143750, 5025, 105, 1, 28398906250, 6802812500

Offset: 1

Wolfdieter Lang

a(n,1) = A034687(n). a(n,m)=: s2(6; n,m), a member of a sequence of unsigned triangles including s2(2; n,m)= A007318(n-1,m-1) (Pascal's triangle). s2(3; n,m)= A035324(n,m), s2(4; n,m)= A035529(n,m), s2(5; n,m)= A048882(n,m).

			{1}; {15,1}; {275,30,1}; {5500,775,45,1}; ...

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

Cf. A039746.

a[n_, m_] := Coefficient[Series[((-1 + (1 - 25*x)^(-1/5))/5)^m, {x, 0, n}], x^n];
Flatten[Table[a[n, m], {n, 1, 9}, {m, 1, n}]][[1 ;; 38]]
(* Jean-François Alcover, Jun 21 2011, after g.f. *)

a(n, m) = 5*(5*(n-1)+m)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n

G.f. for m-th column: ((-1+(1-25*x)^(-1/5))/5)^m.

A008548 Quintuple factorial numbers: Product_{k=0..n-1} (5*k+1).

Original entry on oeis.org

1, 1, 6, 66, 1056, 22176, 576576, 17873856, 643458816, 26381811456, 1213563326976, 61891729675776, 3465936861843456, 211422148572450816, 13953861805781753856, 990724188210504523776, 75295038303998343806976, 6098898102623865848365056, 524505236825652462959394816

Offset: 0

Joe Keane (jgk(AT)jgk.org)

a(n), n>=1, enumerates increasing sextic (6-ary) trees with n vertices. - Wolfdieter Lang, Sep 14 2007

Hankel transform is A169620. - Paul Barry, Dec 03 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..300 (first 50 terms from T. D. Noe)
Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 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 preprint arXiv:1403.5962 [math.CO], 2014-2020.
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), Article 10.6.7, Table 6.3.

Cf. A001147, A007559, A007696, A016861, A034687, A034688, A052562, A047055, A049385, A051150.

List([0..20], n-> Product([0..n], k-> 5*k+1)); # G. C. Greubel, Aug 16 2019

[(&*[5*k+1: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 16 2019

a := n -> mul(5*k+1, k=0..n-1);
G(x):=(1-5*x)^(-1/5): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..16); # Zerinvary Lajos, Apr 03 2009
H := hypergeom([1, 1/5], [], 5*x):
seq(coeff(series(H,x,20),x,n),n=0..16); # Peter Luschny, Oct 08 2015

Table[Product[5k+1,{k,0,n-1}],{n,0,20}]  (* Harvey P. Dale, Apr 23 2011 *)
FoldList[Times,1,NestList[#+5&,1,20]] (* Ray Chandler, Apr 23 2011 *)
FoldList[Times,1,5Range[0, 25] + 1] (* Vincenzo Librandi, Jun 10 2013 *)

x='x+O('x^33); Vec(serlaplace((1-5*x)^(-1/5))) \\ Joerg Arndt, Apr 24 2011

vector(20, n, n--; prod(k=0, n-1, 5*k+1)) \\ Altug Alkan, Oct 08 2015

[product(5*k+1 for k in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 16 2019

a(n) = A049385(n, 1) (first column of triangle).
E.g.f.: (1-5*x)^(-1/5).
a(n) ~ 2^(1/2)*Pi^(1/2)*gamma(1/5)^-1*n^(-3/10)*5^n*e^-n*n^n*{1 + 1/300*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = Sum_{k=0..n} (-5)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005
G.f.: 1/(1-x/(1-5x/(1-6x/(1-10x/(1-11x/(1-15x/(1-16x/(1-20x/(1-21x/(1-25x/(1-.../(1-A008851(n+1)*x/(1-... (continued fraction). - Paul Barry, Dec 03 2009
a(n)=(-4)^n*Sum_{k=0..n} (5/4)^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/Q(0) where Q(k) = 1 - x*(5*k+1)/(1 - x*(5*k+5)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
G.f.: G(0)/2, where G(k)= 1  + 1/(1 - (5*k+1)*x/((5*k+1)*x + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
a(n) = (10n-18)*a(n-2) + (5n-6)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 12 2013
Let T(x) = 1/(1 - 4*x)^(1/4) be the e.g.f. for the sequence of triple factorial numbers A007696. Then the e.g.f. A(x) for the quintuple factorial numbers satisfies T( Integral_{t = 0..x} A(t) dt ) = A(x). Cf. A007559 and A007696. - Peter Bala, Jan 02 2015
O.g.f.: hypergeom([1, 1/5], [], 5*x). - Peter Luschny, Oct 08 2015
a(n) = 5^n * Gamma(n + 1/5) / Gamma(1/5). - Artur Jasinski, Aug 23 2016
D-finite with recurrence: a(n) +(-5*n+4)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/5^4)^(1/5)*(Gamma(1/5) - Gamma(1/5, 1/5)). - Amiram Eldar, Dec 19 2022

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

Original entry on oeis.org

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

A025750 5th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 10, 150, 2625, 49875, 997500, 20662500, 439078125, 9513359375, 209293906250, 4661546093750, 104884787109375, 2380077861328125, 54401779687500000, 1251240932812500000, 28934946571289062500, 672311993862304687500, 15687279856787109375000, 367412607172119140625000

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. A034687, A049393, A340722.

CoefficientList[Series[(6-(1-25x)^(1/5))/5,{x,0,20}],x] (* Harvey P. Dale, Dec 06 2012 *)
a[0] = 1; a[n_] := ((-5)^(n - 1)*Sum[5^(n - k)*StirlingS1[n, k], {k, 1, n}])/n!; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 19 2013, after Vladimir Kruchinin *)
a[n_] := 25^(n-1) * Pochhammer[4/5, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

a(n):=if n=0 then 1 else (sum((-1)^(n-k-1)*binomial(n+k-1,n-1)*sum(2^j*binomial(k,j)*sum(binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i),i,j,n-k+j-1),j,0,k),k,0,n-1))/(n); /* Vladimir Kruchinin, Dec 10 2011 */

a(n):=if n=0 then 1 else -binomial(1/5,n)*(-25)^n/5; /* Tani Akinari, Sep 17 2015 */

G.f.: (6-(1-25*x)^(1/5))/5.
a(n) = 5^(n-1)*4*A034301(n-1)/n!, n >= 2, where 4*A034301(n-1) = (5*n-6)(!^5) = Product_{j=2..n} (5*j-6). - Wolfdieter Lang
a(n) = Sum_{k=0..n-1} (-1)^(n-k-1)*binomial(n+k-1,n-1) * Sum_{j=0..k} 2^j*binomial(k,j) * Sum_{i=j..n-k+j-1} binomial(j,i-j)*binomial(k-j,n-3*(k-j)-i-1)*5^(3*(k-j)+i)/n, n > 0, a(0) = 1. - Vladimir Kruchinin, Dec 10 2011
a(n) = ((-5)^(n-1)*Sum_{k=1..n} (5)^(n-k)*stirling1(n,k))/n!, n>0, a(0) = 1. - Vladimir Kruchinin, Mar 19 2013
From Karol A. Penson, Feb 05 2025: (Start)
a(n) without the initial 1 (i.e., a(n) for n >= 1) is given by
a(n+1) = 5^(2*n)*gamma(n + 4/5)/(gamma(4/5)*(n + 1)!), n >= 0.
a(n+1) = Integral_{x=0..25} x^n*W(x) dx, n >= 0,
  where W(x) = sin(Pi/5)*5^(2/5)*(1 - x/25)^(1/5)/(5*Pi*x^(1/5)). W(x) is a positive  function on x = (0, 25), is singular at x = 0 with the singularity (x)^(-1/5), and it goes to zero at x = 25. (End)
a(n) ~ 25^(n-1) / (Gamma(4/5) * n^(6/5)). - Amiram Eldar, Aug 20 2025

A034789 Related to sextic factorial numbers A008542.

Original entry on oeis.org

1, 21, 546, 15561, 466830, 14471730, 458960580, 14801478705, 483514971030, 15955994043990, 530899438190940, 17785131179396490, 599222112044281740, 20287948650642110340, 689790254121831751560, 23539092421907508521985, 805867752326480585870310, 27668126163209166781547310

Offset: 1

Wolfdieter Lang

Convolution of A004993(n-1) with A025751(n), n >= 1.

Michael De Vlieger, Table of n, a(n) for n = 1..645
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. A004993, A008542, A025751, A034687, A175379.

List([1..20], n-> 6^(n-1)*Product([1..n], j-> 6*j-5)/Factorial(n) ); # G. C. Greubel, Nov 11 2019

[6^(n-1)*(&*[6*j-5: j in [1..n]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( 6^(n-1)*mul(6*j-5, j=1..n)/n!, n=1..20); # G. C. Greubel, Nov 11 2019

Rest@ CoefficientList[Series[(-1 + (1 - 36 x)^(-1/6))/6, {x, 0, 16}], x] (* Michael De Vlieger, Oct 13 2019 *)
Table[6^(2*n-1)*Pochhammer[1/6, n]/n!, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, 6^(n-1)*prod(j=1,n, 6*j-5)/n! ) \\ G. C. Greubel, Nov 11 2019

[6^(n-1)*product( (6*j-5) for j in (1..n))/factorial(n) for n in (1..20)] # G. C. Greubel, Nov 11 2019

a(n) = 6^(n-1)*A008542(n)/n!.
G.f.: (-1+(1-36*x)^(-1/6))/6.
D-finite with recurrence: n*a(n) + 6*(-6*n+5)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 6^(2*n-1) * n^(-5/6) / Gamma(1/6). - Amiram Eldar, Aug 18 2025

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A049375 A convolution triangle of numbers obtained from A034687.

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A008548 Quintuple factorial numbers: Product_{k=0..n-1} (5*k+1).

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

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A025750 5th-order Patalan numbers (generalization of Catalan numbers).

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A034789 Related to sextic factorial numbers A008542.

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