A034300 -id:A034300 - cp's OEIS frontend

A051689 a(n) = (5n+8)(!^5)/8(!^5), related to A034300 ((5n+3)(!^5) quintic, or 5-factorials).

1, 13, 234, 5382, 150696, 4972968, 188972784, 8125829712, 390039826176, 20672110787328, 1198982425665024, 75535892816896512, 5136440711548962816, 374960171943074285568, 29246893411559794274304

Offset: 0

Wolfdieter Lang

Row m=8 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..352

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(13/5))); [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!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(13/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(13/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+8)(!^5))/8(!^5) = A034300(n+2)/8.

E.g.f.: 1/(1-5*x)^(13/5).

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

Original entry on oeis.org

1, 2, 2, 4, 12, 4, 8, 52, 52, 8, 16, 196, 416, 196, 16, 32, 684, 2644, 2644, 684, 32, 64, 2276, 14680, 26440, 14680, 2276, 64, 128, 7340, 74652, 220280, 220280, 74652, 7340, 128, 256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172, 256, 512, 72076, 1637860, 10978444, 27227908, 27227908, 10978444, 1637860, 72076, 512

Offset: 0

Dale Gerdemann, Apr 12 2015

Related triangles may be found by varying the function f(x). If f(x) is a linear function, it can be parameterized as f(x) = a*x + b. With different values for a and b, the following triangles are obtained:
  a\b 1.......2.......3.......4.......5.......6
  -1  A144431
  0   A007318 A038208 A038221
  1   A008292 A256890 A257180 A257606 A257607
  2   A060187 A257609 A257611 A257613 A257615
  3   A142458 A257610 A257620 A257622 A257624 A257626
  4   A142459 A257612 A257621
  5   A142460 A257614 A257623
  6   A142461 A257616 A257625
  7   A142462 A257617 A257627
  8   A167884 A257618
  9   A257608 A257619
The row sums of these, and similarly constructed number triangles, are shown in the following table:
  a\b 1.......2.......3.......4.......5.......6.......7.......8.......9
  0   A000079 A000302 A000400
  1   A000142 A001715 A001725 A049388 A049198
  2   A000165 A002866 A002866 A051580 A051582
  3   A008544 A051578 A037559 A051605 A051607 A051609
  4   A001813 A047053 A000407 A034177 A051618 A051620 A051622
  5   A047055 A008546 A008548 A034300 A034325 A051688 A051690
  6   A047657 A049308 A047058 A034689 A034724 A034788 A053101 A053103
  7   A084947 A144827 A049209 A045754 A034830 A034832 A034834 A053105
  8   A084948 A144828 A147626 A051189 A034908 A034910 A034912 A034976 A053115
  9   A084949 A144829 A147630 A049211 A045756 A035013 A035018 A035021 A035023
  10                                  A051262 A035265 A035273         A035277
  11                                  A254322
  12                                          A145448
The formula can be further generalized to: t(n,m) = f(m+s)*t(n-1,m) + f(n-s)*t(n,m-1), where f(x) = a*x + b. The following table specifies triangles with nonzero values for s (given after the slash).
  a\b  0           1           2          3
  -2   A130595/1
  -1
  0
  1    A110555/-1  A120434/-1  A144697/1  A144699/2
With the absolute value, f(x) = |x|, one obtains A038221/3, A038234/4, A038247/5, A038260/6, A038273/7, A038286/8, A038299/9 (with value for s after the slash).
If f(x) = A000045(x) (Fibonacci) and s = 1, the result is A010048 (Fibonomial).
In the notation of Carlitz and Scoville, this is the triangle of generalized Eulerian numbers A(r, s | alpha, beta) with alpha = beta = 2. Also the array A(2,1,4) in the notation of Hwang et al. (see page 31). - Peter Bala, Dec 27 2019

			Array, t(n, k), begins as:
   1,    2,      4,        8,        16,         32,          64, ...;
   2,   12,     52,      196,       684,       2276,        7340, ...;
   4,   52,    416,     2644,     14680,      74652,      357328, ...;
   8,  196,   2644,    26440,    220280,    1623964,    10978444, ...;
  16,  684,  14680,   220280,   2643360,   27227908,   251195000, ...;
  32, 2276,  74652,  1623964,  27227908,  381190712,  4677894984, ...;
  64, 7340, 357328, 10978444, 251195000, 4677894984, 74846319744, ...;
Triangle, T(n, k), begins as:
    1;
    2,     2;
    4,    12,      4;
    8,    52,     52,       8;
   16,   196,    416,     196,      16;
   32,   684,   2644,    2644,     684,      32;
   64,  2276,  14680,   26440,   14680,    2276,     64;
  128,  7340,  74652,  220280,  220280,   74652,   7340,   128;
  256, 23172, 357328, 1623964, 2643360, 1623964, 357328, 23172,   256;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
L. Carlitz and R. Scoville, Generalized Eulerian numbers: combinatorial applications, J. für die reine und angewandte Mathematik, 265 (1974): 110-37. See Section 3.
Dale Gerdemann, A256890, Plot of t(m,n) mod k , YouTube, 2015.
Hsien-Kuei Hwang, Hua-Huai Chern, and Guan-Huei Duh, An asymptotic distribution theory for Eulerian recurrences with applications, arXiv:1807.01412 [math.CO], 2018-2019.

Cf. A000079, A001715, A008292, A038208, A257180, A257606, A257607, A257609.

Cf. A257610, A257612, A257614, A257616, A257617, A257618, A257619.

A256890:= func< n,k | (&+[(-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n: j in [0..k]]) >;
[A256890(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Oct 18 2022

Table[Sum[(-1)^(k-j)*Binomial[j+3, j] Binomial[n+4, k-j] (j+2)^n, {j,0,k}], {n,0, 9}, {k,0,n}]//Flatten (* Michael De Vlieger, Dec 27 2019 *)

t(n,m) = if ((n<0) || (m<0), 0, if ((n==0) && (m==0), 1, (m+2)*t(n-1, m) + (n+2)*t(n, m-1)));
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n-k, k), ", ");); print(););} \\ Michel Marcus, Apr 14 2015

def A256890(n,k): return sum((-1)^(k-j)*Binomial(j+3,j)*Binomial(n+4,k-j)*(j+2)^n for j in range(k+1))
flatten([[A256890(n,k) for k in range(n+1)] for n in range(11)]) # G. C. Greubel, Oct 18 2022

T(n,k) = t(n-k, k); t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0 else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.
Sum_{k=0..n} T(n, k) = A001715(n).
T(n,k) = Sum_{j = 0..k} (-1)^(k-j)*binomial(j+3,j)*binomial(n+4,k-j)*(j+2)^n. - Peter Bala, Dec 27 2019
Modified rule of Pascal: T(0,0) = 1, T(n,k) = 0 if k < 0 or k > n else T(n,k) = f(n-k) * T(n-1,k-1) + f(k) * T(n-1,k), where f(x) = x + 2. - Georg Fischer, Nov 11 2021
From G. C. Greubel, Oct 18 2022: (Start)
T(n, n-k) = T(n, k).
T(n, 0) = A000079(n). (End)

A034301 a(n) = n-th quintic factorial number divided by 4.

Original entry on oeis.org

1, 9, 126, 2394, 57456, 1666224, 56651616, 2209413024, 97214173056, 4763494479744, 257228701906176, 15176493412464384, 971295578397720576, 67019394909442719744, 4959435223298761261056, 391795382640602139623424, 32910812141810579728367616, 2929062280621141595824717824

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A008546, A008548, A034300, A025750.

List([1..20], n-> Product([1..n], k-> 5*k-1)/4 ); # G. C. Greubel, Aug 23 2019

[&*[5*k-1: k in [1..n]]/4: n in [1..20]]; // G. C. Greubel, Aug 23 2019

a:= n-> mul(5*k-1, k=1..n)/4: seq(a(n), n=1..20); # G. C. Greubel, Aug 23 2019

Table[-5^(n+1)*Pochhammer[-1/5, n+1]/4, {n,20}] (* G. C. Greubel, Aug 23 2019 *)

a(n) = prod(k=1,n, 5*k-1)/4;
vector(20, n, a(n)) \\ G. C. Greubel, Aug 23 2019

[-5^(n+1)*rising_factorial(-1/5, n+1)/4 for n in (1..20)] # G. C. Greubel, Aug 23 2019

a(n) = A008546(n)/4.
4*a(n) = (5*n-1)(!^5) = Product_{j=1..n} (5*j-1).
a(n) = (5*n)!/(5^n*n!*A008548(n)*2*A034323(n)*3*A034300(n)).
E.g.f.: (-1 + (1-5*x)^(-4/5))/4, a(0) = 0.
a(n) ~ sqrt(2*Pi) * 5/(4*Gamma(4/5)) * n^(13/10) * (5*n/e)^n * (1 + (241/300)/n + ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
D-finite with recurrence: a(n) +(-5*n+1)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
Sum_{n>=1} 1/a(n) = 4*(e/5)^(1/5)*(Gamma(4/5) - Gamma(4/5, 1/5)). - Amiram Eldar, Dec 19 2022

Terms a(17) onward added by G. C. Greubel, Aug 23 2019

A034323 a(n) = n-th quintic factorial number divided by 2.

Original entry on oeis.org

1, 7, 84, 1428, 31416, 848232, 27143424, 1004306688, 42180880896, 1982501402112, 103090072909824, 5876134155859968, 364320317663318016, 24409461283442307072, 1757481212407846109184, 135326053355404150407168, 11096736375143140333387776, 965416064637453209004736512

Offset: 1

Wolfdieter Lang

Robert Israel, Table of n, a(n) for n = 1..355
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014-2020.

Cf. A008548, A034300, A034301, A034325, A047055.

a:=[1];; for n in [2..20] do a[n]:=(5*n-3)*a[n-1]; od; a; # G. C. Greubel, Feb 10 2019

[(&*[5*j-3: j in [1..n]])/2: n in [1..20]]; // G. C. Greubel, Feb 10 2019

f:= gfun:-rectoproc({a(n)=(5*n-3)*a(n-1),a(1)=1},a(n),remember):
map(f, [$1..40]); # Robert Israel, Feb 10 2019

Table[Product[5j-3,{j,n}]/2,{n,20}] (* Harvey P. Dale, Nov 25 2013 *)

vector(20, n, prod(j=1, n, 5*j-3)/2) \\ G. C. Greubel, Feb 10 2019

[product(5*j-3 for j in (1..n))/2 for n in (1..20)] # G. C. Greubel, Feb 10 2019

2*a(n) = (5*n-3)(!^5) = Product_{j=1..n} (5*j-3).
E.g.f.: (-1 + (1-5*x)^(-2/5))/2, with a(0) = 0.
a(n) ~ sqrt(2*Pi) * 5/(2*Gamma(2/5)) * n^(9/10) * (5*n/e)^n * (1 + (109/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001
D-finite with recurrence a(n) = (5*n-3)*a(n-1). - Robert Israel, Feb 10 2019
From Amiram Eldar, Dec 19 2022: (Start)
a(n) = A047055(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/5^3)^(1/5)*(Gamma(2/5) - Gamma(2/5, 1/5)). (End)

A034325 a(n) is the n-th quintic factorial number divided by 5.

Original entry on oeis.org

1, 10, 150, 3000, 75000, 2250000, 78750000, 3150000000, 141750000000, 7087500000000, 389812500000000, 23388750000000000, 1520268750000000000, 106418812500000000000, 7981410937500000000000, 638512875000000000000000

Offset: 1

Wolfdieter Lang

Michael De Vlieger, Table of n, a(n) for n = 1..354
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 594.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A008548, A034300, A034301, A034323, A052562.

List([1..20], n-> 5^(n-1)*Factorial(n) ); # G. C. Greubel, Aug 23 2019

[5^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 23 2019

seq(5^(n-1)*n!, n=1..20); # G. C. Greubel, Aug 23 2019

Array[5^(# - 1) #! &, 16] (* Michael De Vlieger, May 30 2019 *)

vector(20, n, 5^(n-1)*n!) \\ G. C. Greubel, Aug 23 2019

[5^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 23 2019

5*a(n) = (5*n)(!^5) = Product_{j=1..n} 5*j = 5^(n-1)*n!.
E.g.f.: (-1 + (1-5*x)^(-1))/5, a(0) = 0.
D-finite with recurrence: a(n) - 5*n*a(n-1) = 0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 5*(exp(1/5)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*(1-exp(-1/5)). (End)

A035275 One sixth of deca-factorial numbers.

Original entry on oeis.org

1, 16, 416, 14976, 688896, 38578176, 2546159616, 193508130816, 16641699250176, 1597603128016896, 169345931569790976, 19644128062095753216, 2475160135824064905216, 336621778472072827109376, 49146779656922632757968896, 7666897626479930710243147776

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..320
Index entries for sequences related to factorial numbers.

Cf. A034300, A035272, A035273, A035274, A035275, A035276, A035277, A035278, A035279, A045757.

List([1..20], n-> Product([1..n], j-> 10*j-4)/6 ); # G. C. Greubel, Nov 11 2019

[(&*[10*j-4: j in [1..n]])/6: n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(10*j-4, j=1..n)/6, n=1..20); # G. C. Greubel, Nov 11 2019

Table[10^n*Pochhammer[6/10, n]/6, {n, 20}] (* G. C. Greubel, Nov 11 2019 *)

vector(20, n, prod(j=1,n, 10*j-4)/6 ) \\ G. C. Greubel, Nov 11 2019

[product( (10*j-4) for j in (1..n))/6 for n in (1..20)] # G. C. Greubel, Nov 11 2019

6*a(n) = (10*n-4)(!^10) = Product_{j=1..n} (10*j-4).
a(n) = 2^n*3*A034300(n) where 3*A034300(n) = (5*n-2)(!^5).
E.g.f.: (-1 + (1-10*x)^(-3/5))/6.
a(n) = (Pochhammer(6/10,n) * 10^n)/6.
Sum_{n>=1} 1/a(n) = 6*(e/10^4)^(1/10)*(Gamma(3/5) - Gamma(3/5, 1/10)). - Amiram Eldar, Dec 22 2022

A051687 a(n) = (5n+6)(!^5)/6, related to A008548 ((5n+1)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 11, 176, 3696, 96096, 2978976, 107243136, 4396968576, 202260554496, 10315288279296, 577656143640576, 35237024762075136, 2325643634296958976, 165120698035084087296, 12549173050666390634496

Offset: 0

Wolfdieter Lang

Row m=6 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..353

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(11/5))); [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, 10, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(11/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(11/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+6)(!^5))/6(!^5).

E.g.f.: 1/(1-5*x)^(11/5).

A051691 a(n) = (5n+10)(!^5)/10(!^5), related to A052562 ((5n)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 15, 300, 7500, 225000, 7875000, 315000000, 14175000000, 708750000000, 38981250000000, 2338875000000000, 152026875000000000, 10641881250000000000, 798141093750000000000, 63851287500000000000000

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..352

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(15/5))); [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!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(15/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(15/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+10)(!^5))/10(!^5) = A052562(n+2)/(5*10).

E.g.f.: 1/(1-5*x)^3.

A051688 a(n) = (5n+7)(!^5)/7(!^5), related to A034323 ((5n+2)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 12, 204, 4488, 121176, 3877632, 143472384, 6025840128, 283214486016, 14727153272832, 839447736551424, 52045759666188288, 3487065897634615296, 251068744629692301312, 19332293336486307201024

Offset: 0

Wolfdieter Lang

Row m=7 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..353

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(12/5))); [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, 11, 5!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(12/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-5*x)^(12/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+7)(!^5))/7(!^5) = A034323(n+2)/7.

E.g.f.: 1/(1-5*x)^(12/5).

A051690 a(n) = (5n+9)(!^5)/9(!^5), related to A034301 ((5n+2)(!^5) quintic, or 5-factorials).

Original entry on oeis.org

1, 14, 266, 6384, 185136, 6294624, 245490336, 10801574784, 529277164416, 28580966878464, 1686277045829376, 107921730933080064, 7446599434382524416, 551048358144306806784, 43532820293400237735936, 3656756904645619969818624, 325451364513460177313857536

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(6; m,n) := ((5*n+m)(!^5))/m(!^5), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..352

Cf. A052562, A008548(n+1), A034323(n+1), A034300(n+1), A034301(n+1), A034325(n+1), A051687-A051691 (rows m=0..10).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-5*x)^(14/5))); [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!, 5}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 5*x)^(14/5), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

my(x='x+O('x^30)); Vec(serlaplace(1/(1-5*x)^(14/5))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((5*n+9)(!^5))/9(!^5) = A034301(n+2)/9.

E.g.f.: 1/(1-5*x)^(14/5).

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A051689 a(n) = (5*n+8)(!^5)/8(!^5), related to A034300 ((5*n+3)(!^5) quintic, or 5-factorials).

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A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 2.

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A034301 a(n) = n-th quintic factorial number divided by 4.

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A034323 a(n) = n-th quintic factorial number divided by 2.

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A034325 a(n) is the n-th quintic factorial number divided by 5.

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A035275 One sixth of deca-factorial numbers.

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A051689 a(n) = (5n+8)(!^5)/8(!^5), related to A034300 ((5n+3)(!^5) quintic, or 5-factorials).

A256890 Triangle T(n,k) = t(n-k, k); t(n,m) = f(m)t(n-1,m) + f(n)t(n,m-1), where f(x) = x + 2.

A051687 a(n) = (5n+6)(!^5)/6, related to A008548 ((5n+1)(!^5) quintic, or 5-factorials).

A051691 a(n) = (5n+10)(!^5)/10(!^5), related to A052562 ((5n)(!^5) quintic, or 5-factorials).

A051688 a(n) = (5n+7)(!^5)/7(!^5), related to A034323 ((5n+2)(!^5) quintic, or 5-factorials).

A051690 a(n) = (5n+9)(!^5)/9(!^5), related to A034301 ((5n+2)(!^5) quintic, or 5-factorials).