A034910 -id:A034910 - cp's OEIS frontend

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.

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)

A090802 Triangle read by rows: a(n,k) = number of k-length walks in the Hasse diagram of a Boolean algebra of order n.

Original entry on oeis.org

1, 2, 1, 4, 4, 2, 8, 12, 12, 6, 16, 32, 48, 48, 24, 32, 80, 160, 240, 240, 120, 64, 192, 480, 960, 1440, 1440, 720, 128, 448, 1344, 3360, 6720, 10080, 10080, 5040, 256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320

Offset: 0

Ross La Haye, Feb 10 2004

Row sums = A010842(n); Row sums from column 1 on = A066534(n) = n*A010842(n-1) = A010842(n) - 2^n.
a(n,k) = n! = k! = A000142(n) for n = k; a(n,n-1) = 2*n! = A052849(n) for n > 1; a(n,n-2) = 2*n! = A052849(n) for n > 2; a(n,n-3) = (4/3)*n! = A082569(n) for n > 3; a(n,n-1)/a(2,1) = n!/2! = A001710(n) for n > 1; a(n,n-2)/ a(3,1) = n!/3! = A001715(n) for n > 2; a(n,n-3)/a(4,1) = n!/4! = A001720(n) for n > 3.
a(2k, k) = A052714(k+1). a(2k-1, k) = A034910(k).
a(n,0) = A000079(n); a(n,1) = A001787(n) = row sums of A003506; a(n,2) = A001815(n) = 2!*A001788(n-1); a(n,3) = A052771(n) = 3!*A001789(n); a(n,4) = A052796(n) = 4!*A003472(n); ceiling[a(n,1) / 2] = A057711(n); a(n,5) = 5!*A054849(n).
In a class of n students, the number of committees (of any size) that contain an ordered k-sized subcommittee is a(n,k). - Ross La Haye, Apr 17 2006
Antidiagonal sums [1,2,5,12,30,76,198,528,1448,4080,...] appear to be binomial transform of A000522 interleaved with itself, i.e., 1,1,2,2,5,5,16,16,65,65,... - Ross La Haye, Sep 09 2006
Let P(A) be the power set of an n-element set A. Then a(n,k) = the number of ways to add k elements of A to each element x of P(A) where the k elements are not elements of x and order of addition is important. - Ross La Haye, Nov 19 2007
The derivatives of x^n evaluated at x=2. - T. D. Noe, Apr 21 2011

			{1};
{2, 1};
{4, 4, 2};
{8, 12, 12, 6};
{16, 32, 48, 48, 24};
{32, 80, 160, 240, 240, 120};
{64, 192, 480, 960, 1440, 1440, 720};
{128, 448, 1344, 3360, 6720, 10080, 10080, 5040};
{256, 1024, 3584, 10752, 26880, 53760, 80640, 80640, 40320}
a(5,3) = 240 because P(5,3) = 60, 2^(5-3) = 4 and 60 * 4 = 240.

T. D. Noe, Rows n = 0..100, flattened
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Eric Weisstein, Walk
Eric Weisstein, Boolean Algebra
Eric Weisstein, Hasse Diagram

Cf. A000142, A001710, A001715, A001720, A001787, A001788, A001789, A001815, A003472, A010842, A052771, A052796, A052849, A054849, A057711, A066534, A082569.

Cf. A038207, A007318.

Flatten[Table[n!/(n-k)! * 2^(n-k), {n, 0, 8}, {k, 0, n}]] (* Ross La Haye, Feb 10 2004 *)

a(n, k) = 0 for n < k. a(n, k) = k!*C(n, k)*2^(n-k) = P(n, k)*2^(n-k) = (2n)!!/((n-k)!*2^k) = k!*A038207(n, k) = A068424*2^(n-k) = Sum[C(n, m)*P(n-m, k), {m, 0, n-k}] = Sum[C(n, n-m)*P(n-m, k), {m, 0, n-k}] = n!*Sum[1/(m!*(n-m-k)!), {m, 0, n-k}] = k!*Sum[C(n, m)*C(n-m, k), {m, 0, n-k}] = k!*Sum[C(n, n-m)*C(n-m, k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, n-m-k), {m, 0, n-k}] = k!*C(n, k)*Sum[C(n-k, m), {m, 0, n-k}] for n >= k.
a(n, k) = 0 for n < k. a(n, k) = n*a(n-1, k-1) for n >= k >= 1.
E.g.f. (by columns): exp(2x)*x^k.

More terms from Ray Chandler, Feb 26 2004

Entry revised by Ross La Haye, Aug 18 2006

A034909 One third of octo-factorial numbers.

Original entry on oeis.org

1, 11, 209, 5643, 197505, 8492715, 433128465, 25554579435, 1712156822145, 128411761660875, 10658176217852625, 969894035824588875, 96019509546634298625, 10274087521489869952875, 1181520064971335044580625, 145326967991474210483416875, 19037832806883121573327610625

Offset: 1

Wolfdieter Lang

Index entries for sequences related to factorial numbers.

Cf. A045755, A034908, A034909, A034910, A034911, A034912, A144756.

3*a(n) = (8*n-5)(!^8) = product(8*j-5, j=1..n).
E.g.f.: (-1+(1-8*x)^(-3/8))/3.
G.f.: x/(1-11x/(1-8x/(1-19x/(1-16x/(1-27x/(1-24x/(1-35x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A144756(n)/3.
Sum_{n>=1} 1/a(n) = 3*(e/8^5)^(1/8)*(Gamma(3/8) - Gamma(3/8, 1/8)). (End)

A034912 One sixth of octo-factorial numbers.

Original entry on oeis.org

1, 14, 308, 9240, 351120, 16151520, 872182080, 54075288960, 3785270227200, 295251077721600, 25391592684057600, 2386809712301414400, 243454590654744268800, 26780004972021869568000, 3160040586698580609024000, 398165113924021156737024000, 53354125265818835002761216000

Offset: 1

Wolfdieter Lang

Robert Israel, Table of n, a(n) for n = 1..333
Index entries for sequences related to factorial numbers.

Cf. A034176, A034908, A034909, A034910, A034911, A045755, A147626.

[n le 1 select 1 else (8*n-2)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 20 2022

f:= proc(n) option remember; procname(n-1)*(8*n-2) end proc:
f(1):= 1:
map(f,[$1..20]); # Robert Israel, Mar 20 2018

Table[8^n*Pochhammer[3/4,n]/6, {n,40}] (* G. C. Greubel, Oct 20 2022 *)

[8^n*rising_factorial(3/4,n)/6 for n in range(1,40)] # G. C. Greubel, Oct 20 2022

6*a(n) = (8*n-2)(!^8) = Product_{j=1..n} (8*j - 2) = 2^n*3*A034176(n), where 3*A034176(n) = (4*n-1)(!^4) = Product_{j=1..n} (4*j - 1).
E.g.f.: (-1+(1-8*x)^(-3/4))/6.
G.f.: x/(1-14*x/(1-8*x/(1-22*x/(1-16*x/(1-30*x/(1-24*x/(1-38*x/(1-32*x/(1-...(continued fraction). - Philippe Deléham, Jan 07 2012
From G. C. Greubel, Oct 20 2022: (Start)
a(n) = (1/6) * 8^n * Pochhammer(n, 3/4).
a(n) = 2*(4*n - 1)*a(n-1). (End)
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A147626(n+1)/6.
Sum_{n>=1} 1/a(n) = 6*(e/8^2)^(1/8)*(Gamma(3/4) - Gamma(3/4, 1/8)). (End)

A034911 One fifth of octo-factorial numbers.

Original entry on oeis.org

1, 13, 273, 7917, 292929, 13181805, 698635665, 42616775565, 2940557513985, 226422928576845, 19245948929031825, 1789873250399959725, 180777198290395932225, 19704714613653156612525, 2305451609797419323665425, 288181451224677415458178125, 38328133012882096255937690625

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. A034908, A034909, A034910, A034911, A034912, A045755, A147625.

[n le 1 select 1 else (8*n-3)*Self(n-1): n in [1..40]]; // G. C. Greubel, Oct 20 2022

With[{nn=20},CoefficientList[Series[(-1+(1-8x)^(-5/8))/5,{x,0,nn}],x] Range[0,nn]!] (* or *) FoldList[Times,Range[5,200,8]]/5 (* Harvey P. Dale, May 25 2016 *)

[8^n*rising_factorial(5/8,n)/5 for n in range(1,40)] # G. C. Greubel, Oct 20 2022

5*a(n) = (8*n-3)(!^8) = Product_{j=1..n} 8*j-3.
E.g.f.: (-1+(1-8*x)^(-5/8))/5.
G.f.: x/(1-13*x/(1-8*x/(1-21*x/(1-16*x/(1-29*x/(1-24*x/(1-37*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
D-finite with recurrence: a(n) = (8*n-3)*a(n-1). - R. J. Mathar, Jan 28 2020
a(n) = (1/5)* 8^n * Pochhammer(n, 5/8). - G. C. Greubel, Oct 20 2022
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A147625(n+1)/5.
Sum_{n>=1} 1/a(n) = 5*(e/8^3)^(1/8)*(Gamma(5/8) - Gamma(5/8, 1/8)). (End)

A034975 One seventh of octo-factorial numbers.

Original entry on oeis.org

1, 15, 345, 10695, 417105, 19603935, 1078216425, 67927634775, 4822862069025, 381006103452975, 33147531000408825, 3149015445038838375, 324348590839000352625, 36002693583129039141375, 4284320536392355657823625, 544108708121829168543600375, 73454675596446937753386050625

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. A007696, A034176, A034908, A034909, A034910, A034911, A034912, A034976, A045755, A049210.

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

Table[8^n*Pochhammer[7/8, n]/7, {n, 40}] (* G. C. Greubel, Oct 21 2022 *)

[8^n*rising_factorial(7/8,n)/7 for n in range(1,40)] # G. C. Greubel, Oct 21 2022

7*a(n) = (8*n-1)!^8 = Product_{j=1..n} (8*j-1) = (8*n)!/((2*n)!*2^(6*n)*3^2*5 * A045755(n)*A007696(n)*A034909(n)*A034911(n)*A034176(n)).
E.g.f.: (-1+(1-8*x)^(-7/8))/7.
G.f.: x/(1-15*x/(1-8*x/(1-23*x/(1-16*x/(1-31*x/(1-24*x/(1-39*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (1/7) * 8^n * Pochhammer(n, 7/8). - G. C. Greubel, Oct 21 2022
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A049210(n)/7.
Sum_{n>=1} 1/a(n) = 7*(e/8)^(1/8)*(Gamma(7/8) - Gamma(7/8, 1/8)). (End)

A034976 One eighth of octo-factorial numbers.

Original entry on oeis.org

1, 16, 384, 12288, 491520, 23592960, 1321205760, 84557168640, 6088116142080, 487049291366400, 42860337640243200, 4114592413463347200, 427917611000188108800, 47926772432021068185600, 5751212691842528182272000, 736155224555843607330816000

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. A034908, A034909, A034910, A034911, A034912, A034975, A045755, A051189.

[8^(n-1)*Factorial(n): n in [1..40]]; // G. C. Greubel, Oct 20 2022

Table[8^(n-1)*n!, {n,40}] (* G. C. Greubel, Oct 20 2022 *)

[8^(n-1)*factorial(n) for n in range(1,40)] # G. C. Greubel, Oct 20 2022

8*a(n) = (8*n)!^8 = Product_{j=1..n} 8*j = 8^n*n!.
E.g.f.: (-1+(1-8*x)^(-1))/8.
G.f.: x/(1-16*x/(1-8*x/(1-24*x/(1-16*x/(1-32*x/(1-24*x/(1-40*x/(1-32*x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 8*(exp(1/8)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 8*(1-exp(-1/8)). (End)

A203516 a(n) = Product_{1 <= i < j <= n} 2*(i+j-1).

Original entry on oeis.org

1, 4, 192, 184320, 4954521600, 4794391461888000, 204135216112950312960000, 451965950843675288237663846400000, 60040562704967329457107799785403842560000000, 542366306792798635131534558788357929673196306432000000000

Offset: 1

Clark Kimberling, Jan 03 2012

nonn

Each term divides its successor, as in A034910.

See A093883 for a guide to related sequences.

G. C. Greubel, Table of n, a(n) for n = 1..33

Cf. A000178, A005408, A034910, A093883, A203516, A203517.

[2^Binomial(n,2)*(&*[Factorial(2*k)/Factorial(k): k in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Feb 19 2024

a:= n-> mul(mul(2*(i+j-1), i=1..j-1), j=2..n):
seq(a(n), n=1..12);  # Alois P. Heinz, Jul 23 2017

f[j_] := 2 j - 1; z = 15;
v[n_] := Product[Product[f[k] + f[j], {j, 1, k - 1}], {k, 2, n}]
d[n_] := Product[(i - 1)!, {i, 1, n}]    (* A000178 *)
Table[v[n], {n, 1, z}]                   (* A203516 *)
Table[v[n + 1]/(4 v[n]), {n, 1, z - 1}]  (* A034910 *)
Table[v[n]/d[n], {n, 1, 20}]             (* A203517 *)
Table[2^(-1/24 - 3*n/2 + 3*n^2/2) * Glaisher^(3/2) * Pi^(1/4 - n/2) * BarnesG[1/2 + n]/E^(1/8), {n, 1, 10}] (* Vaclav Kotesovec, Sep 01 2023 *)

a(n) = my(pd=1); for(j=1, n, for(i=1, j-1, pd=pd*2*(i+j-1))); pd \\ Felix Fröhlich, Jul 23 2017

[2^binomial(n,2)*product(factorial(2*k)/factorial(k) for k in range(n)) for n in range(1,21)] # G. C. Greubel, Feb 19 2024

a(n) ~ sqrt(A) * 2^(-7/24 - n + 3*n^2/2) * exp(-1/24 + n/2 - 3*n^2/4) * n^(1/24 - n/2 + n^2/2), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 01 2023
From G. C. Greubel, Feb 19 2024: (Start)
a(n) = BarnesG(n+1)*A203517(n).
a(n) = 2^binomial(n,2) * Product_{j=1..n-1} (2j)!/j!. (End)

Name edited by Alois P. Heinz, Jul 23 2017

A254619 a(n) = 4^n(2n + 1)!/n!.

Original entry on oeis.org

1, 24, 960, 53760, 3870720, 340623360, 35424829440, 4250979532800, 578133216460800, 87876248902041600, 14763209815542988800, 2716430606059909939200, 543286121211981987840000, 117349802181788109373440000

Offset: 0

Peter Bala, Feb 03 2015

Cf. A002391, A034910, A073000, A254381, A254620.

Maple
```
seq(4^n*(2*n + 1)!/n!, n = 0..13);
```

Table[4^n (2n+1)!/n!,{n,0,20}] (* Harvey P. Dale, Oct 02 2021 *)

E.g.f.: 1/(1 - 16*x)^(3/2) = 1 + 24*x + 960*x^2/2! + 53760*x^3/3! + ....
Recurrence equation: a(n) = 8*(2*n + 1)*a(n-1) with a(0) = 1.
2nd order recurrence equation: a(n) = (20*n + 6)*a(n-1) - 16*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 24.
Define a sequence b(n) := a(n)*sum {k = 0..n} 1/((2*k + 1)*4^k) beginning [1, 26, 1052, 59032, 4251984, 374204832, 38917967808, ...]. It is not difficult to check that b(n) also satisfies the previous 2nd order recurrence equation (and so is an integer sequence). From this observation we can obtain the continued fraction expansion
log(3) = Sum {k >= 0} 1/((2*k + 1)*4^k) = 1 + 2/(24 - 16*3^2/(46 - 16*5^2/(66 - ... - 16*(2*n - 1)^2/((20*n + 6) - ... )))).
Alternative 2nd order recurrence equation: a(n) = (12*n + 10)*a(n-1) + 16*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 24.
Define now a sequence c(n) := a(n)*sum {k = 0..n} (-1)^k/((2*k + 1)*4^k) beginning [1, 22, 892, 49832, 3589584, 315853152, 32849393088, ...], which, along with a(n), satisfies the alternative 2nd order recurrence equation. From this observation we find the continued fraction expansion 2*arctan(1/2) = Sum {k >= 0} (-1)^k/((2*k + 1)*4^k) = 1 - 2/(24 + 16*3^2/(34 + 16*5^2/(46 + ... + 16*(2*n - 1)^2/((12*n + 10) + ... )))). Cf. A254381 and A254620.

A254381 a(n) = 3^n(2n + 1)!/n!.

Original entry on oeis.org

1, 18, 540, 22680, 1224720, 80831520, 6304858560, 567437270400, 57878601580800, 6598160580211200, 831368233106611200, 114728816168712345600, 17209322425306851840000, 2787910232899709998080000, 485096380524549539665920000, 90227926777566214377861120000

Offset: 0

Peter Bala, Feb 04 2015

Cf. A002391, A034910, A073000, A093766, A254619, A254620.

Maple
```
seq(3^n*(2*n + 1)!/n!, n = 0..13);
```

Table[3^n(2n + 1)!/n!, {n, 0, 19}] (* Alonso del Arte, Feb 04 2015 *)

E.g.f.: 1/(1 - 12*x)^(3/2) = 1 + 18*x + 540*x^2/2! + 22680*x^3/3! + ....
Recurrence equation: a(n) = 6*(2*n + 1)*a(n-1) with a(0) = 1.
2nd order recurrence equation: a(n) = 8*(n + 1)*a(n-1) + 12*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 18.
Define a sequence b(n) := a(n)*sum {k = 0..n} (-1)^k/((2*k + 1)*3^k) beginning [1, 16, 492, 20544, 1111056, 73299456, 5718022848, ...]. It is not difficult to check that b(n) also satisfies the previous 2nd order recurrence equation (and so is an integer sequence). Using this observation we obtain the continued fraction expansion Pi/(2*sqrt(3)) = Sum {k >= 0} (-1)^k/( (2*k + 1)*3^k ) = 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). Cf. A254619 and A254620.

cp's OEIS Frontend

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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A090802 Triangle read by rows: a(n,k) = number of k-length walks in the Hasse diagram of a Boolean algebra of order n.

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A034909 One third of octo-factorial numbers.

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A034912 One sixth of octo-factorial numbers.

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A034911 One fifth of octo-factorial numbers.

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A034975 One seventh of octo-factorial numbers.

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A034976 One eighth of octo-factorial numbers.

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A203516 a(n) = Product_{1 <= i < j <= n} 2*(i+j-1).

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.

A254619 a(n) = 4^n(2n + 1)!/n!.

A254381 a(n) = 3^n(2n + 1)!/n!.