A034787 -id:A034787 - 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

A008543 Sextuple factorial numbers: Product_{k=0..n-1} (6*k + 5).

Original entry on oeis.org

1, 5, 55, 935, 21505, 623645, 21827575, 894930575, 42061737025, 2229272062325, 131527051677175, 8549258359016375, 606997343490162625, 46738795448742522125, 3879320022245629336375, 345259481979861010937375, 32799650788086796039050625, 3312764729596766399944113125

Offset: 0

Joe Keane (jgk(AT)jgk.org)

Vincenzo Librandi, Table of n, a(n) for n = 0..300
Wolfdieter Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), Article 00.2.4.

a(n) = A013988(n+1, 1) (first column of triangle).

Cf. A004994, A034787, A049308, A047058, A048994, A051151.

[Round(6^n*Gamma(n+5/6)/Gamma(5/6)): n in [0..20]]; // G. C. Greubel, Dec 03 2019

Maple
```
f := n->product( (6*k-1),k=0..n);
```

FoldList[Times,1,6Range[0,15]+5]  (* Harvey P. Dale, Feb 20 2011 *)
Table[6^n*Pochhammer[5/6, n], {n, 0, 20}] (* G. C. Greubel, Dec 03 2019 *)
CoefficientList[Series[(1 - 6x)^(-5/6), {x, 0, 20}], x] Range[0, 20]! (* Nikolaos Pantelidis, Jan 31 2023 *)

a(n)=prod(k=1,n,6*k-1) \\ Charles R Greathouse IV, Aug 17 2011

[6^n*rising_factorial(5/6, n) for n in (0..20)] # G. C. Greubel, Dec 03 2019

a(n) = 5*A034787(n) = (6*n-1)(!^6), n >= 1, a(0) := 1.
E.g.f.: (1 - 6*x)^(-5/6).
a(n) ~ 2^(1/2)*Pi^(1/2)*Gamma(5/6)^-1*n^(1/3)*6^n*e^-n*n^n*(1 + (1/72)*n^-1 + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
G.f.: 1/(1-5x/(1-6x/(1-11x/(1-12x/(1-17x/(1-18x/(1-23x/(1-24x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-1)^n*Sum_{k=0..n} 6^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 - 1/Q(0))/x 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) = 6^n * Gamma(n+5/6) / Gamma(5/6). - Vaclav Kotesovec, Jan 28 2015
D-finite with recurrence: a(n) +(-6*n+1)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
From Nikolaos Pantelidis, May 22 2022: (Start)
G.f.: 1/G(0), where G(k) = 1 - (12*k+5)*x - 6*(k+1)*(6*k+5)*x^2/G(k+1) (a continued fraction);
which starts 1/(1-5*x-30*x^2/(1-17*x-132*x^2/(1-29*x-306*x^2/(1-41*x-552*x^2/(1-53*x-870*x^2/(1-65*x-1260*x^2/(1-...))))))) (a Jacobi continued fraction).
(End)
Sum_{n>=0} 1/a(n) = 1 + (e/6)^(1/6)*(Gamma(5/6) - Gamma(5/6, 1/6)). - Amiram Eldar, Dec 18 2022

A034689 a(n) = n-th sextic factorial number divided by 2.

Original entry on oeis.org

1, 8, 112, 2240, 58240, 1863680, 70819840, 3116072960, 155803648000, 8725004288000, 540950265856000, 36784618078208000, 2722061737787392000, 217764939022991360000, 18727784755977256960000, 1722956197549907640320000, 168849707359890948751360000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..325
Index entries for sequences related to factorial numbers.

Cf. A007559, A008542, A034723, A034724, A034787, A034788, A047657, A047058.

[n le 1 select 1 else (6*n-4)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 21 2022

Table[6^n*Pochhammer[1/3, n]/2, {n, 40}] (* G. C. Greubel, Oct 21 2022 *)

[6^n*rising_factorial(1/3,n)/2 for n in range(1,40)] # G. C. Greubel, Oct 21 2022

2*a(n) = (6*n-4)(!^6) = Product_{j=1..n} (6*j-4) = 2^n*A007559(n), A007559(n) = (3*n-2)(!^3) = Product_{j=1..n} (3*j-2).
E.g.f.: (-1 + (1-6*x)^(-1/3))/2.
D-finite with recurrence: a(n) = 2*(3*n-2)*a(n-1). - R. J. Mathar, Feb 24 2020
a(n) = 3*6^(n-1)*Pochhammer(n, 1/3). - G. C. Greubel, Oct 21 2022
From Amiram Eldar, Dec 18 2022: (Start)
a(n) = A047657(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/6^4)^(1/6)*(Gamma(1/3, 1/6) - Gamma(1/3)). (End)

A034788 a(n) is the n-th sextic factorial number divided by 6.

Original entry on oeis.org

1, 12, 216, 5184, 155520, 5598720, 235146240, 11287019520, 609499054080, 36569943244800, 2413616254156800, 173780370299289600, 13554868883344588800, 1138608986200945459200, 102474808758085091328000, 9837581640776168767488000, 1003433327359169214283776000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..345

Cf. A008542, A034689, A034723, A034724, A034787.

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

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

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

Table[6^(n-1)*n!,{n,20}] (* Harvey P. Dale, Dec 22 2013 *)

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

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

6*a(n) = (6*n)(!^6) = Product_{j=1..n} 6*j = 6^n*n!.
E.g.f.: (-1 + 1/(1-6*x))/6.
D-finite with recurrence: a(n) - 6*n*a(n-1) = 0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 6*(exp(1/6)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(1-exp(-1/6)). (End)

A053101 a(n) = ((6n+8)(!^6))/8(!^6), related to A034689 (((6n+2)(!^6))/2 sextic, or 6-factorials).

Original entry on oeis.org

1, 14, 280, 7280, 232960, 8852480, 389509120, 19475456000, 1090625536000, 67618783232000, 4598077259776000, 340257717223424000, 27220617377873920000, 2340973094497157120000, 215369524693738455040000

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..344

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, this sequence, A053102, A053103 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(7/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(7/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(7/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((6*n+8)(!^6))/8(!^6)= A034689(n+2)/8.

E.g.f.: 1/(1-6*x)^(7/3).

A053103 a(n) = ((6n+10)(!^6))/10(!^6), related to A034724 (((6n+4)(!^6))/4 sextic, or 6-factorials).

Original entry on oeis.org

1, 16, 352, 9856, 335104, 13404160, 616591360, 32062750720, 1859639541760, 119016930672640, 8331185147084800, 633170071178444800, 51919945836632473600, 4568955233623657676800, 429481791960623821619200

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..343

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, A053101, A053102, this sequence (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(8/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 16 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 15, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(16/6), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 16 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(8/3))) \\ G. C. Greubel, Aug 16 2018

a(n) = ((6*n+10)(!^6))/10(!^6) = A034724(n+2)/10.

E.g.f.: 1/(1-6*x)^(8/3).

A053100 a(n) = ((6n+7)(!^6))/7, related to A008542 ((6n+1)(!^6) sextic, or 6-factorials).

Original entry on oeis.org

1, 13, 247, 6175, 191425, 7082725, 304557175, 14923301575, 820781586625, 50067676784125, 3354534344536375, 244881007151155375, 19345599564941274625, 1644375963020008343125, 149638212634820759224375

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..344

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), this sequence, A053101, A053102, A053103 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(13/6))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 12, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn=20},CoefficientList[Series[1/(1-6x)^(13/6),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Apr 20 2015 *)

x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(13/6))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((6*n+7)(!^6))/7(!^6) = A008542(n+2)/7.

E.g.f.: 1/(1-6*x)^(13/6).

A053102 a(n) = ((6n+9)(!^6))/9(!^6), related to A034723 (((6n+3)(!^6))/3 sextic, or 6-factorials).

Original entry on oeis.org

1, 15, 315, 8505, 280665, 10945935, 492567075, 25120920825, 1431892487025, 90209226682575, 6224436641097675, 466832748082325625, 37813452594668375625, 3289770375736148679375, 305948644943461827181875

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(7; m,n) := ((6*n+m)(!^6))/m(!^6), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..344

Cf. A047058, A008542(n+1), A034689(n+1), A034723(n+1), A034724(n+1), A034787(n+1), A034788(n+1), A053100, A053101, this sequence, A053103 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-6*x)^(15/6))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 14, 5!, 6}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 6*x)^(15/6), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-6*x)^(15/6))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((6*n+9)(!^6))/9(!^6) = A034723(n+2)/9.

E.g.f.: 1/(1-6*x)^(15/6).

A025751 6th-order Patalan numbers (generalization of Catalan numbers).

Original entry on oeis.org

1, 1, 15, 330, 8415, 232254, 6735366, 202060980, 6213375135, 194685754230, 6191006984514, 199237861137996, 6475230486984870, 212188322111965740, 7002214629694869420, 232473525705869664744, 7758803920433400060831, 260148131449825766745510, 8758320425477467480432170

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. A034787, A203145.

CoefficientList[Series[(7 - (1 - 36*x)^(1/6))/6, {x, 0, 20}], x] (* Vincenzo Librandi, Dec 29 2012 *)
a[n_] := 36^(n-1) * Pochhammer[5/6, n-1]/n!; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Aug 20 2025 *)

a[0]:1$ a[1]:1$ a[n]:=(6/n)*(6*n-7)*a[n-1]$ makelist(a[n],n,0,1000); /* Tani Akinari, Aug 03 2014 */

G.f.: (7-(1-36*x)^(1/6))/6.
a(n) = 6^(n-1)*5*A034787(n-1)/n!, n >= 2, where 5*A034787(n-1)=(6*n-7)(!^6) = Product_{j=2..n} (6*j - 7). - Wolfdieter Lang.
a(n) ~ 36^(n-1) / (Gamma(5/6) * n^(7/6)). - Amiram Eldar, Aug 20 2025

A049224 A convolution triangle of numbers obtained from A025751.

Original entry on oeis.org

1, 15, 1, 330, 30, 1, 8415, 885, 45, 1, 232254, 26730, 1665, 60, 1, 6735366, 825858, 58320, 2670, 75, 1, 202060980, 25992252, 2003562, 106560, 3900, 90, 1, 6213375135, 830282805, 68351283, 4038741, 174825, 5355, 105, 1, 194685754230

Offset: 1

Wolfdieter Lang

a(n,1) = A025751(n); a(n,1)= 6^(n-1)*5*A034787(n-1)/n!, n >= 2.

G.f. for m-th column: ((1-(1-36*x)^(1/6))/6)^m.

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

Cf. A048966, A049223. Row sums = A025759.

T(n,m):=(m*sum(binomial(-m+2*i-1,i-1)*2^(2*n-2*i)*sum(binomial(k,n-k-i)*3^(k+i-m)*(-1)^(n-k-i)*binomial(n+k-1,n-1),k,0,n-i),i,m,n))/n; /* Vladimir Kruchinin, Dec 21 2011 */

a(n, m) = 6*(6*(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.: [(1-(1-36*x)^(1/6))/6]^m=sum(n>=m, T(n,m)*x^n), T(n,m)=(m*sum(i=m..n, binomial(-m+2*i-1,i-1)*2^(2*n-2*i)*sum(k=0..n-i, binomial(k,n-k-i)*3^(k+i-m)*(-1)^(n-k-i)*binomial(n+k-1,n-1))))/n. - Vladimir Kruchinin, Dec 21 2011

Showing 1-10 of 10 results.

cp's OEIS Frontend

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

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A008543 Sextuple factorial numbers: Product_{k=0..n-1} (6*k + 5).

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A034689 a(n) = n-th sextic factorial number divided by 2.

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A034788 a(n) is the n-th sextic factorial number divided by 6.

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A053101 a(n) = ((6*n+8)(!^6))/8(!^6), related to A034689 (((6*n+2)(!^6))/2 sextic, or 6-factorials).

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A053103 a(n) = ((6*n+10)(!^6))/10(!^6), related to A034724 (((6*n+4)(!^6))/4 sextic, or 6-factorials).

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A053100 a(n) = ((6*n+7)(!^6))/7, related to A008542 ((6*n+1)(!^6) sextic, or 6-factorials).

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A053101 a(n) = ((6n+8)(!^6))/8(!^6), related to A034689 (((6n+2)(!^6))/2 sextic, or 6-factorials).

A053103 a(n) = ((6n+10)(!^6))/10(!^6), related to A034724 (((6n+4)(!^6))/4 sextic, or 6-factorials).

A053100 a(n) = ((6n+7)(!^6))/7, related to A008542 ((6n+1)(!^6) sextic, or 6-factorials).

A053102 a(n) = ((6n+9)(!^6))/9(!^6), related to A034723 (((6n+3)(!^6))/3 sextic, or 6-factorials).