A004993 -id:A004993 - cp's OEIS frontend

A008542 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+1).

1, 1, 7, 91, 1729, 43225, 1339975, 49579075, 2131900225, 104463111025, 5745471106375, 350473737488875, 23481740411754625, 1714167050058087625, 135419196954588922375, 11510631741140058401875, 1047467488443745314570625, 101604346379043295513350625

Offset: 0

Joe Keane (jgk(AT)jgk.org)

nonn

a(n), n>=1, enumerates increasing heptic (7-ary) trees with n vertices. - Wolfdieter Lang, Sep 14 2007; see a D. Callan comment on A007559 (number of increasing quarterny trees).

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Index entries for sequences related to factorial numbers.

Cf. A085158, A034689, A034723, A034724, A034787, A034788, A004993, A047058, A047657, A051151.

Cf. k-fold factorials: A000142, A001147 (and A000165, A006882), A007559 (and A032031, A008544, A007661), A007696 (and A001813, A008545, A047053, A007662), A008548 (and A052562, A047055, A085157), A045754 (and A084947, A114799), A045755.

List([0..20], n-> Product([0..n-1], k-> (6*k+1) )); # G. C. Greubel, Aug 17 2019

[1] cat [(&*[(6*k+1): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Aug 17 2019

a := n -> mul(6*k+1, k=0..n-1);
G(x):=(1-6*x)^(-1/6): 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..15); # Zerinvary Lajos, Apr 03 2009

Table[Product[(6*k+1), {k,0,n-1}], {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008, modified by G. C. Greubel, Aug 17 2019 *)
FoldList[Times, 1, 6Range[0, 20] + 1] (* Vincenzo Librandi, Jun 10 2013 *)
Table[6^n*Pochhammer[1/6, n], {n,0,20}] (* G. C. Greubel, Aug 17 2019 *)

a(n)=prod(k=1,n-1,6*k+1) \\ Charles R Greathouse IV, Jul 19 2011

[product((6*k+1) for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 17 2019

E.g.f.: (1-6*x)^(-1/6).
a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(1/6)^-1*n^(-1/3)*6^n*e^-n*n^n*{1 + 1/72*n^-1 - ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = Sum_{k=0..n} (-6)^(n-k)*A048994(n, k). - Philippe Deléham, Oct 29 2005
G.f.: 1+x/(1-7x/(1-6x/(1-13x/(1-12x/(1-19x/(1-18x/(1-25x/(1-24x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-5)^n*Sum_{k=0..n} (6/5)^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*(6*k+1)/(1 - x*(6*k+6)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
a(n) = A085158(6*n-5). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-6*n+5)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/6^5)^(1/6)*(Gamma(1/6) - Gamma(1/6, 1/6)). - Amiram Eldar, Dec 18 2022

A034835 Expansion of 1/(1-49*x)^(1/7); related to sept-factorial numbers A045754.

Original entry on oeis.org

1, 7, 196, 6860, 264110, 10722866, 450360372, 19365495996, 847240449825, 37560993275575, 1682732498745760, 76028913806967520, 3459315578217022160, 158330213003009860400, 7283189798138453578400, 336483368673996555322080, 15604416222256590253061460, 726064307753233111186565580

Offset: 0

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 0..445
Armin Straub, Victor H. Moll, and Tewodros Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009), 31-41, eq (1.10).
Index entries for sequences related to factorial numbers.

Cf. A045754, A004993, A034829, A034830, A034831, A034832, A034833, A034834, A220086.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!(1/(1 - 49*x)^(1/7))); // G. C. Greubel, Feb 22 2018

CoefficientList[Series[1/(1 - 49*x)^(1/7), {x,0,50}], x] (* G. C. Greubel, Feb 22 2018 *)

my(x='x+O('x^30)); Vec(1/(1 - 49*x)^(1/7)) \\ G. C. Greubel, Feb 22 2018

a(n) = 7^n*A045754(n)/n!, n >= 1, where A045754(n) = (7*n-6)(!^7) = Product_{j=1..n} (7*j-6).
G.f.: (1-49*x)^(-1/7).
D-finite with recurrence: n*a(n) + 7*(-7*n+6)*a(n-1) = 0. - R. J. Mathar, Jan 28 2020
a(n) ~ 7^(2*n) * n^(-6/7) / Gamma(1/7). - Amiram Eldar, Aug 18 2025

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

Original entry on oeis.org

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)

A004996 a(n) = 6^n/n! * Product_{k=0..n-1} (6*k - 1).

Original entry on oeis.org

1, -6, -90, -1980, -50490, -1393524, -40412196, -1212365880, -37280250810, -1168114525380, -37146041907084, -1195427166827976, -38851382921909220, -1273129932671794440, -42013287778169216520

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A004993.

List([0..20], n-> 6^n*Product([0..n-1], k-> 6*k-1)/Factorial(n) ); # G. C. Greubel, Aug 20 2019

[1] cat [6^n*(&*[6*k-1: k in [0..n-1]])/Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 20 2019

seq(6^n*product(6*k-1, k = 0..n-1)/n!, n = 0..20); # G. C. Greubel, Aug 20 2019

Table[6^(2*n)*Pochhammer[-1/6, n]/n!, {n,0,20}] (* G. C. Greubel, Aug 20 2019 *)

vector(20, n, n--; 6^n*prod(j=0,n-1, 6*j-1)/n! ) \\ G. C. Greubel, Aug 20 2019

[6^(2*n)*rising_factorial(-1/6, n)/factorial(n) for n in (0..20)] # G. C. Greubel, Aug 20 2019

G.f.: (1 - 36*x)^(1/6).
a(n) ~ -1/6*Gamma(5/6)^-1*n^(-7/6)*6^(2*n)*{1 + 7/72*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
a(n) = 6^(2*n) * binomial(n-7/6, n). - G. C. Greubel, Aug 20 2019
D-finite with recurrence: n*a(n) +6*(-6*n+7)*a(n-1)=0. - R. J. Mathar, Jan 17 2020

A034977 Expansion of 1/(1-64*x)^(1/8), related to octo-factorial numbers A045755.

Original entry on oeis.org

1, 8, 288, 13056, 652800, 34467840, 1884241920, 105517547520, 6014500208640, 347504456499200, 20294260259553280, 1195516422562775040, 70933974405391319040, 4234212626044897198080, 254052757562693831884800, 15310912855778348268257280, 926310227774590070229565440

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..500
Armin Straub, Victor H. Moll, and Tewodros Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009), 31-41, eq. (1.10).
Index entries for sequences related to factorial numbers.

Cf. A004993, A034855, A045755, A203142.

[n le 1 select 8^(n-1) else 8*(8*n-15)*Self(n-1)/(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022

CoefficientList[Series[1/(1-64x)^(1/8),{x,0,30}],x] (* Harvey P. Dale, May 20 2011 *)

[2^(6*n)*rising_factorial(1/8,n)/factorial(n) for n in range(40)] # G. C. Greubel, Oct 21 2022

a(n) = 8^n*A045755(n)/n!, n >= 1, where A045755(n) = (8*n-7)!^8 = Product_{j=1..n} (8*j-7).
G.f.: (1-64*x)^(-1/8).
D-finite with recurrence: n*a(n) = 8*(8*n-7)*a(n-1). - R. J. Mathar, Jan 28 2020
a(n) ~ 2^(6*n) * n^(-7/8) / Gamma(1/8). - Amiram Eldar, Aug 18 2025

a(11) corrected by Harvey P. Dale, May 20 2011

A224881 Expansion of 1/(1 - 16*x)^(1/8).

Original entry on oeis.org

1, 2, 18, 204, 2550, 33660, 460020, 6440280, 91773990, 1325624300, 19354114780, 285033326760, 4227994346940, 63094684869720, 946420273045800, 14259398780556720, 215673406555920390, 3273161111260438860, 49824785804742235980, 760483572809223601800

Offset: 0

Paul D. Hanna, Jul 23 2013

nonn

			G.f.: A(x) = 1 + 2*x + 18*x^2 + 204*x^3 + 2550*x^4 + 33660*x^5 + ...
where
A(x)^8 = 1 + 16*x + 256*x^2 + 4096*x^3 + 65536*x^4 + ... + 16^n*x^n + ...
Also,
A(x)^4 = 1 + 8*x + 96*x^2 + 1280*x^3 + 17920*x^4 + 258048*x^5 + ... + 4^n*A000984(n)*x^n + ...
A(x)^2 = 1 + 4*x + 40*x^2 + 480*x^3 + 6240*x^4 + 84864*x^5 + ... + 2^n*A004981(n)*x^n + ...

Seiichi Manyama, Table of n, a(n) for n = 0..500

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), this sequence (b=16), A034688 (b=25), A298799 (b=27), A004993 (b=36), A034835 (b=49).

Cf. A301271.

List([0..20],n->(2^n/Factorial(n))*Product([0..n-1],k->8*k+1)); # Muniru A Asiru, Jun 23 2018

seq(coeff(series(1/(1-16*x)^(1/8), x,50),x,n+1),n=0..20); # Muniru A Asiru, Jun 23 2018

CoefficientList[Series[1/(1-16*x)^(1/8), {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 24 2013 *)

{a(n)=polcoeff(1/(1-16*x +x*O(x^n))^(1/8),n)}
for(n=0,30,print1(a(n),", "))

{a(n)=(2^n/n!)*prod(k=0,n-1,8*k + 1)}
for(n=0,30,print1(a(n),", "))

a(n) = (2^n/n!) * Product_{k=0..n-1} (8*k + 1).

a(n) ~ 16^n/(GAMMA(1/8)*n^(7/8)). - Vaclav Kotesovec, Jul 24 2013

A298799 Expansion of (1-27*x)^(-1/9).

Original entry on oeis.org

1, 3, 45, 855, 17955, 398601, 9167823, 216098685, 5186368440, 126201632040, 3104560148184, 77049538223112, 1926238455577800, 48452305767226200, 1225151160114148200, 31118839466899364280, 793530406405933789140, 20305042752151835192700

Offset: 0

Seiichi Manyama, Jun 22 2018

nonn

Conjecture: a(p*n) == a(n) (mod p^2) for prime p == 1 (mod 9) and all positive integers n except those n of the form n = m*p + k for 0 <= m <= (p-1)/9 and 1 <= k <= (p-1)/9. Cf. A034171, A004981 and A004982. - Peter Bala, Dec 23 2019

Seiichi Manyama, Table of n, a(n) for n = 0..700

(1-b*x)^(-1/A003557(b)): A000984 (b=4), A004981 (b=8), A004987 (b=9), A098658 (b=12), A224881 (b=16), A034688 (b=25), this sequence (b=27), A004993 (b=36), A034835 (b=49).

Cf. A305991, A034171, A004981, A004982.

List([0..20],n->(3^n/Factorial(n))*Product([0..n-1],k->9*k+1)); # Muniru A Asiru, Jun 23 2018

seq(coeff(series((1-27*x)^(-1/9), x, n+1), x, n), n=0..20); # Muniru A Asiru, Jun 23 2018
# Alternative:
A298799 := n -> (-27)^n*binomial(-1/9, n):
seq(A298799(n), n=0..17); # Peter Luschny, Dec 26 2019

N=20; x='x+O('x^N); Vec((1-27*x)^(-1/9))

a(n) = 3^n/n! * Product_{k=0..n-1} (9*k + 1) for n > 0.
a(n) ~ 3^(3*n) / (Gamma(1/9) * n^(8/9)). - Vaclav Kotesovec, Jun 23 2018
From Peter Luschny, Dec 26 2019: (Start)
a(n) = (-27)^n*binomial(-1/9, n).
a(n) = n! * [x^n] hypergeom([1/9], [1], 27*x). (End)
D-finite with recurrence: n*a(n) +3*(-9*n+8)*a(n-1)=0. - R. J. Mathar, Jan 20 2020

A035308 Expansion of 1/(1-100*x)^(1/10), related to deca-factorial numbers A045757.

Original entry on oeis.org

1, 10, 550, 38500, 2983750, 244667500, 20796737500, 1812287125000, 160840482343750, 14475643410937500, 1317283550395312500, 120950580536296875000, 11187928699607460937500, 1041337978963463671875000, 97439482317295529296875000, 9159311337825779753906250000

Offset: 0

Wolfdieter Lang

Seiichi Manyama, Table of n, a(n) for n = 0..500
Armin Straub, Victor H. Moll, and Tewodros Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arith. 140 (1) (2009), 31-41, eq (1.10).
Index entries for sequences related to factorial numbers.

Cf. A045757, A035024, A004993, A000984, A004987, A256191.

CoefficientList[Series[1/(1-100*x)^(1/10), {x, 0, 20}], x] (* Amiram Eldar, Aug 18 2025 *)

a(n) = 10^n*A045757(n)/n!, n >= 1, where A045757(n) = (10*n-9)(!^10) = Product_{j=1..n} (10*j-9).
G.f.: (1-100*x)^(-1/10).
a(n) ~ 10^(2*n) * n^(-9/10) / Gamma(1/10). - Amiram Eldar, Aug 18 2025

A248328 Square array read by antidiagonals downwards: super Patalan numbers of order 6.

Original entry on oeis.org

1, 6, 30, 126, 90, 990, 3276, 1260, 1980, 33660, 93366, 24570, 20790, 50490, 1161270, 2800980, 560196, 324324, 424116, 1393524, 40412196, 86830380, 14004900, 6162156, 5513508, 9754668, 40412196, 1414426860, 2753763480, 372130200, 132046200, 89791416, 108694872, 242473176, 1212365880

Offset: 0

Tom Richardson, Oct 04 2014

Generalization of super Catalan numbers, A068555, based on Patalan numbers of order 6, A025751.

			T(0..4,0..4) is
  1          6         126       3276      93366
  30         90        1260      24570     560196
  990        1980      20790     324324    6162156
  33660      50490     424116    5513508   89791416
  1161270    1393524   9754668   108694872 1548901926

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, A025751, A004993 (first row), A004994 (first column), A004995 (second row), A004996 (second column), A248324, A248325, A248326, A248329, A248332.

matrix(5, 5, nn, kk, n=nn-1;k=kk-1;(-1)^k*36^(n+k)*binomial(n-1/6,n+k)) \\ Michel Marcus, Oct 09 2014

T(0,0)=1, T(n,k) = T(n-1,k)*(36*n-6)/(n+k), T(n,k) = T(n,k-1)*(36*k-30)/(n+k).
G.f.: (x/(1-36*x)^(5/6)+y/(1-36*y)^(1/6))/(x+y-36*x*y).
T(n,k) = (-1)^k*36^(n+k)*binomial(n-1/6,n+k).

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

cp's OEIS Frontend

A008542 Sextuple factorial numbers: Product_{k=0..n-1} (6*k+1).

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A034835 Expansion of 1/(1-49*x)^(1/7); related to sept-factorial numbers A045754.

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

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A004996 a(n) = 6^n/n! * Product_{k=0..n-1} (6*k - 1).

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A034977 Expansion of 1/(1-64*x)^(1/8), related to octo-factorial numbers A045755.

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A224881 Expansion of 1/(1 - 16*x)^(1/8).

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A298799 Expansion of (1-27*x)^(-1/9).

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