A007696 -id:A007696 - cp's OEIS frontend

A045755 8-fold factorials: a(n) = Product_{k=0..n-1} (8*k+1).

1, 1, 9, 153, 3825, 126225, 5175225, 253586025, 14454403425, 939536222625, 68586144251625, 5555477684381625, 494437513909964625, 47960438849266568625, 5035846079172989705625, 569050606946547836735625, 68855123440532288245010625, 8882310923828665183606370625

Offset: 0

Wolfdieter Lang

nonn

G. C. Greubel, Table of n, a(n) for n = 0..330

Cf. k-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542, A045754.

Cf. A048994, A051187, A113135.

List([0..20], n-> Product([0..n-1], j-> 8*j+1) ); # G. C. Greubel, Nov 11 2019

[1] cat [(&*[8*j+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019

Maple
```
a := n->product(8*k+1), k=0..(n-1));
```

Table[8^n*Pochhammer[1/8, n], {n,0,20}] (* G. C. Greubel, Nov 11 2019 *)

PARI
```
a(n)=prod(k=0, n, 8*k+1);
```

[product( (8*j+1) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 11 2019

a(n+1) = (8*n+1)(!^8).
a(n) = Sum_{k=0..n} (-8)^(n-k)*A048994(n, k); A048994 = Stirling-1 numbers.
E.g.f.: (1-8*x)^(-1/8).
G.f.: 1+x/(1-9x/(1-8x/(1-17x/(1-16x/(1-25x/(1-24x/(1-33x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
a(n) = (-7)^n*Sum_{k=0..n} (8/7)^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*(8*k+1)/(1 - x*(8*k+8)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 20 2013
G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 2*x*(8*k+1)/(2*x*(8*k+1) - 1 + 16*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(8*k+1)/(x*(8*k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 05 2013
a(n) = 8^n * Gamma(n + 1/8) / Gamma(1/8). - Artur Jasinski,Aug 23 2016
a(n) ~ sqrt(2*Pi) * 8^n * n^(n - 3/8)/(Gamma(1/8)*exp(n)). - Ilya Gutkovskiy, Sep 10 2016
D-finite with recurrence: a(n) +(-8*n+7)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
Sum_{n>=0} 1/a(n) = 1 + (e/8^7)^(1/8)*(Gamma(1/8) - Gamma(1/8, 1/8)). - Amiram Eldar, Dec 20 2022

Additional comments from Philippe Deléham and Paul D. Hanna, Oct 29 2005

Edited by N. J. A. Sloane, Oct 14 2008 at the suggestion of Artur Jasinski.

A045756 Expansion of e.g.f. (1-9*x)^(-1/9), 9-factorial numbers.

Original entry on oeis.org

1, 1, 10, 190, 5320, 196840, 9054640, 498005200, 31872332800, 2326680294400, 190787784140800, 17361688356812800, 1736168835681280000, 189242403089259520000, 22330603564532623360000, 2835986652695643166720000, 385694184766607470673920000, 55925656791158083247718400000

Offset: 0

Wolfdieter Lang

Nine-fold factorials of numbers 9k+1, k = 0, 1, 2, ... - M. F. Hasler, Feb 14 2020

G. C. Greubel, Table of n, a(n) for n = 0..325 [a(0)=1 inserted by _Georg Fischer_, Feb 15 2020]
Peter Luschny, Mulitfactorials.
Index entries for sequences related to factorial numbers.

Cf. A008542, A048994, A114806 (9-fold factorials), A132393.

Cf. k-fold factorials : A000142 ("1-fold"), A001147 (2-fold), A007559 (3), A007696 (4), A008548 (5), A008542 (6), A045754 (7), A045755 (8), A144773 (10), A256268 (combined table).

List([0..20], n-> Product([0..n-1], j-> 9*j+1) ); # G. C. Greubel, Nov 11 2019

[1] cat [(&*[9*j+1: j in [0..n-1]]): n in [1..20]]; // G. C. Greubel, Nov 11 2019

seq( mul(9*j+1, j=0..n-1), n=0..20); # G. C. Greubel, Nov 11 2019

Table[9^n*Pochhammer[1/9, n], {n,0,20}] (* G. C. Greubel, Nov 11 2019 *)

vector(21, n, prod(j=0,n-2, 9*j+1) ) \\ G. C. Greubel, Nov 11 2019

[product( (9*j+1) for j in (0..n-1)) for n in (0..20)] # G. C. Greubel, Nov 11 2019

a(n+1) = (9*n+1)(!^9) = Product_{k=0..n-1} (9*k+1), n >= 0.
E.g.f. (1-9*x)^(-1/9).
D-finite with recurrence: a(n) +(-9*n+8)*a(n-1)=0. - R. J. Mathar, Jan 17 2020
a(n) = A114806(9n-8). - M. F. Hasler, Feb 14 2020
a(n) = Sum_{k = 0..n} (-9)^(n - k) * A048994(n, k) = Sum_{k = 0..n} 9^(n - k) * A132393(n, k). Philippe Deléham, Sep 20 2008
a(n) = (-8)^n * sum_{k = 0..n} (9/8)^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
a(n) = 9^n * Gamma(n + 1/9) / Gamma(1/9). - Artur Jasinski Aug 23 2016
a(n) ~ sqrt(2 * Pi) * 9^n * n^(n - 7/18)/(Gamma(1/9) * exp(n)). - Ilya Gutkovskiy, Sep 10 2016
Sum_{n>=0} 1/a(n) = 1 + (e/9^8)^(1/9)*(Gamma(1/9) - Gamma(1/9, 1/9)). - Amiram Eldar, Dec 21 2022

a(0)=1 inserted; merged with A144772; formulas and programs changed accordingly by Georg Fischer, Feb 15 2020

A034176 One third of quartic factorial numbers.

Original entry on oeis.org

1, 7, 77, 1155, 21945, 504735, 13627845, 422463195, 14786211825, 576662261175, 24796477230525, 1165434429834675, 59437155921568425, 3269043575686263375, 192873570965489539125, 12151034970825840964875, 814119343045331344646625, 57802473356218525469910375

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..350
Maxie D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), Article 10.6.7, p 39.

Cf. A007696, A034177, A034256, A025749.

a:=[1];; for n in [2..20] do a[n]:=(4*n-1)*a[n-1]; od; a; # G. C. Greubel, Aug 15 2019

[n le 1 select 1 else (4*n-1)*Self(n-1): n in [1..20]]; // G. C. Greubel, Aug 15 2019

A034176:=n->`if`(n=1, 1, (4*n-1)*A034176(n-1)); seq(A034176(n), n=1..20); # G. C. Greubel, Aug 15 2019

Table[4^n*Pochhammer[3/4, n]/3, {n, 20}] (* G. C. Greubel, Aug 15 2019 *)

m=20; v=concat([1], vector(m-1)); for(n=2, m, v[n]=(4*n-1)*v[n-1]); v \\ G. C. Greubel, Aug 15 2019

[4^n*rising_factorial(3/4, n)/3 for n in (1..20)] # G. C. Greubel, Aug 15 2019

3*a(n) = (4*n-1)(!^4) := Product_{j=1..n} 4*j-1 = (4*n-1)!!/A007696(n) = (4*n)!/(4^n*(2*n)!*A007696(n)), A007696(n)=(4*n-3)(!^4), n >= 1;
E.g.f.: (-1 + (1-4*x)^(-3/4))/3.
a(n) ~ 4/3 * 2^(1/2) * Pi^(1/2) * Gamma(3/4)^(-1) * n^(5/4) * 2^(2*n) * e^(-n) * n^n * {1 + 71/96*n^(-1) + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
G.f.: 1/Q(0) where Q(k) = 1 - x + 2*(2*k-1)*x - 4*x*(k+1) / Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 03 2013
D-finite with recurrence: a(n) + (-4*n+1) * a(n-1) = 0. - R. J. Mathar, Feb 24 2020
Sum_{n>=1} 1/a(n) = 3*exp(1/4)*(Gamma(3/4) - Gamma(3/4, 1/4)) / sqrt(2). - Amiram Eldar, Dec 18 2022
a(n) = 4^(n-1) * Gamma(n + 3/4) / Gamma(7/4). - Peter McNair, May 06 2024

A034177 a(n) is the n-th quartic factorial number divided by 4.

Original entry on oeis.org

1, 8, 96, 1536, 30720, 737280, 20643840, 660602880, 23781703680, 951268147200, 41855798476800, 2009078326886400, 104472072998092800, 5850436087893196800, 351026165273591808000, 22465674577509875712000, 1527665871270671548416000, 109991942731488351485952000

Offset: 1

Wolfdieter Lang

			G.f. = x + 8*x^2 + 96*x^3 + 1536*x^4 + 30720*x^5 + 737820*x^6 + ...

Michael De Vlieger, Table of n, a(n) for n = 1..365
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 513.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019.

Cf. A007696, A000407, A034176. First column of triangle A048786.
A052570 is an essentially identical sequence. - Philippe Deléham, Sep 18 2008
Equals the second right hand column of A167569 divided by 2. - Johannes W. Meijer, Nov 12 2009

List([1..20], n-> 4^(n-1)*Factorial(n) ); # G. C. Greubel, Aug 15 2019

[4^(n-1)*Factorial(n): n in [1..20]]; // G. C. Greubel, Aug 15 2019

[seq(n!*4^(n-1), n=1..16)]; # Zerinvary Lajos, Sep 23 2006

Array[4^(# - 1) #! &, 16] (* Michael De Vlieger, May 30 2019 *)

vector(20, n, 4^(n-1)*n!) \\ G. C. Greubel, Aug 15 2019

[4^(n-1)*factorial(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019

4*a(n) = (4*n)(!^4) = Product_{j=1..n} 4*j = 4^n * n!.
E.g.f.: (-1 + 1/(1-4*x))/4.
D-finite with recurrence: a(n) -4*n*a(n-1)=0. - R. J. Mathar, Feb 24 2020
From Amiram Eldar, Jan 08 2022: (Start)
Sum_{n>=1} 1/a(n) = 4*(exp(1/4)-1).
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*(1-exp(-1/4)). (End)

A111146 Triangle T(n,k), read by rows, given by [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 5, 8, 0, 0, 6, 15, 17, 16, 0, 0, 24, 62, 68, 49, 32, 0, 0, 120, 322, 359, 243, 129, 64, 0, 0, 720, 2004, 2308, 1553, 756, 321, 128, 0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256, 0, 0, 40320, 119664

Offset: 0

Philippe Deléham, Oct 19 2005

Let R(m,n,k), 0<=k<=n, the Riordan array (1, x*g(x)) where g(x) is g.f. of the m-fold factorials . Then Sum_{k, 0<=k<=n} = R(m,n,k) = Sum_{k, 0<=k<=n} T(n,k)*m^(n-k).
For m = -1, R(-1,n,k) is A026729(n,k).
For m = 0, R(0,n,k) is A097805(n,k).
For m = 1, R(1,n,k) is A084938(n,k).
For m = 2, R(2,n,k) is A111106(n,k).

			Triangle begins:
.1;
.0, 1;
.0, 0, 2;
.0, 0, 1, 4;
.0, 0, 2, 5, 8;
.0, 0, 6, 15, 17, 16;
.0, 0, 24, 62, 68, 49, 32;
.0, 0, 120, 322, 359, 243, 129, 64;
.0, 0, 720, 2004, 2308, 1553, 756, 321, 128;
.0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256;
.0, 0, 40320, 119664, 148232, 105048, 49840, 18066, 5756, 1793, 512;
....................................................................
At y=2: Sum_{k=0..n} 2^k*T(n,k) = A113327(n) where (1 + 2*x + 8*x^2 + 36*x^3 +...+ A113327(n)*x^n +..) = 1/(1 - 2/1!*x*(1! + 2!*x + 3!*x^2 + 4!*x^3 +..) ).
At y=3: Sum_{k=0..n} 3^k*T(n,k) = A113328(n) where (1 + 3*x + 18*x^2 + 117*x^3 +...+ A113328(n)*x^n +..) = 1/(1 - 3/2!*x*(2! + 3!*x + 4!*x^2 + 5!*x^3 +..) ).
At y=4: Sum_{k=0..n} 4^k*T(n,k) = A113329(n) where (1 + 4*x + 32*x^2 + 272*x^3 +...+ A113329(n)*x^n +..) = 1/(1 - 4/3!*x*(3! + 4!*x + 5!*x^2 + 6!*x^3 +..) ).

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000045, A026729, A051295, A084938, A097805, A111106, A112934.
Cf. m-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542.
Cf. A113326, A113327 (y=2), A113328 (y=3), A113329 (y=4), A113330 (y=5), A113331 (y=6).

T[n_, k_] := Module[{x = X + X*O[X]^n, y = Y + Y*O[Y]^k}, A = 1/(1 - x*y*Sum[x^j*Product[y + i, {i, 0, j - 1}], {j, 0, n}]); Coefficient[ Coefficient[A, X, n], Y, k]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, from PARI *)

{T(n,k)=local(x=X+X*O(X^n),y=Y+Y*O(Y^k)); A=1/(1-x*y*sum(j=0,n,x^j*prod(i=0,j-1,y+i))); return(polcoeff(polcoeff(A,n,X),k,Y))} (Hanna)

Sum_{k, 0<=k<=n} (-1)^(n-k)*T(n, k) = A000045(n+1), Fibonacci numbers.
Sum_{k, 0<=k<=n} T(n, k) = A051295(n).
Sum_{k, 0<=k<=n} 2^(n-k)*T(n, k) = A112934(n).
T(0, 0) = 1, T(n, n) = 2^(n-1).
G.f.: A(x, y) = 1/(1 - x*y*Sum_{j>=0} (y-1+j)!/(y-1)!*x^j ). - Paul D. Hanna, Oct 26 2005

A114799 Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 6, 7, 8, 18, 30, 44, 60, 78, 98, 120, 288, 510, 792, 1140, 1560, 2058, 2640, 6624, 12240, 19800, 29640, 42120, 57624, 76560, 198720, 379440, 633600, 978120, 1432080, 2016840, 2756160, 7352640, 14418720, 24710400, 39124800

Offset: 0

Jonathan Vos Post, Feb 18 2006

Many of the terms yield multifactorial primes a(n) + 1, e.g.: a(2) + 1 = 3, a(4) + 1 = 5, a(6) + 1 = 7, a(9) + 1 = 19, a(10) + 1 = 31, a(12) + 1 = 61, a(13) + 1 = 79, a(24) + 1 = 12241, a(25) + 1 = 19801, a(26) + 1 = 29641, a(29) + 1 = 76561, a(31) + 1 = 379441, a(35) + 1 = 2016841, a(36) + 1 = 2756161, ...

Equivalently, product of all positive integers <= n congruent to n (mod 7). - M. F. Hasler, Feb 23 2018

			a(40) = 40 * a(40-7) = 40 * a(33) = 40 * (33*a(26)) = 40 * 33 * (26*a(19)) = 40 * 33 * 26 * (19*a(12)) = 40 * 33 * 26 * 19 * (12*a(5)) = 40 * 33 * 26 * 19 * 12 5 = 39124800.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Multifactorial.
Index entries for sequences related to factorial numbers

Cf. A045754, A084947, A288094.

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), A085158 (and A008542, A047058, A047657), A045755.

a:= function(n)
    if n<1 then return 1;
    else return n*a(n-7);
    fi;
  end;
List([0..40], n-> a(n) ); # G. C. Greubel, Aug 20 2019

b:= func< n | (n lt 8) select n else n*Self(n-7) >;
[1] cat [b(n): n in [1..40]]; // G. C. Greubel, Aug 20 2019

A114799 := proc(n)
    option remember;
    if n < 1 then
        1;
    else
        n*procname(n-7) ;
    end if;
end proc:
seq(A114799(n),n=0..40) ; # R. J. Mathar, Jun 23 2014
A114799 := n -> product(n-7*k,k=0..(n-1)/7); # M. F. Hasler, Feb 23 2018

a[n_]:= If[n<1, 1, n*a[n-7]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Aug 20 2019 *)

A114799(n,k=7)=prod(j=0,(n-1)\k,n-j*k) \\ M. F. Hasler, Feb 23 2018

def a(n):
    if (n<1): return 1
    else: return n*a(n-7)
[a(n) for n in (0..40)] # G. C. Greubel, Aug 20 2019

a(n) = 1 for n <= 1, else a(n) = n*a(n-7).

Sum_{n>=0} 1/a(n) = A288094. - Amiram Eldar, Nov 10 2020

Edited by M. F. Hasler, Feb 23 2018

A051142 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -4, 1, 32, -12, 1, -384, 176, -24, 1, 6144, -3200, 560, -40, 1, -122880, 70144, -14400, 1360, -60, 1, 2949120, -1806336, 415744, -47040, 2800, -84, 1, -82575360, 53526528, -13447168, 1732864, -125440, 5152, -112, 1, 2642411520, -1795424256, 483835904

Offset: 1

Wolfdieter Lang

a(n,m) = R_n^m(a=0, b=4) in the notation of the given 1961 and 1962 references.
a(n,m) is a Jabotinsky matrix, i.e., the monic row polynomials E(n,x) := Sum_{m=1..n} a(n,m)*x^m = Product_{j=0..n-1} (x - 4*j), n >= 1, and E(0,x) := 1 are exponential convolution polynomials (see A039692 for the definition and a Knuth reference).
This is the signed Stirling1 triangle with diagonal d >= 0 (main diagonal d = 0) scaled with 4^d.
Also the Bell transform of the quadruple factorial numbers Product_{k=0..n-1} (4*k+4) (A047053) giving unsigned values and adding 1, 0, 0, 0, ... as column 0. For the definition of the Bell transform, see A264428 and for cross-references A265606. - Peter Luschny, Dec 31 2015

			Triangle a(n,m) (with rows n >= 1 and columns m = 1..n) begins:
        1;
       -4,     1;
       32,   -12,      1;
     -384,   176,    -24,    1;
     6144, -3200,    560,  -40,   1,
  -122880, 70144, -14400, 1360, -60, 1;
  ...
3rd row o.g.f.: E(3,x) = 32*x - 12*x^2 + x^3.

Richell O. Celeste, Roberto B. Corcino, and Ken Joffaniel M. Gonzales, Two Approaches to Normal Order Coefficients, Journal of Integer Sequences, Vol. 20 (2017), Article 17.3.5.
Wolfdieter Lang, First 10 rows.
D. S. Mitrinovic, Sur une classe de nombres reliés aux nombres de Stirling, Comptes rendus de l'Académie des sciences de Paris, t. 252 (1961), 2354-2356.
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77.
D. S. Mitrinovic and R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz., No. 77 (1962), 1-77 [jstor stable version].

First (m=1) column sequence is: A047053(n-1).
Row sums (signed triangle): A008545(n-1)*(-1)^(n-1).
Row sums (unsigned triangle): A007696(n).
Cf. A008275 (Stirling1 triangle, b=1), A039683 (b=2), A051141 (b=3).
Cf. A039692, A264428, A265606.

Table[StirlingS1[n, m] 4^(n - m), {n, 9}, {m, n}] // Flatten (* Michael De Vlieger, Dec 31 2015 *)

# uses[bell_transform from A264428]
# Unsigned values and an additional first column (1,0,0,0, ...).
def A051142_row(n):
    multifact_4_4 = lambda n: prod(4*k + 4 for k in (0..n-1))
    mfact = [multifact_4_4(k) for k in (0..n)]
    return bell_transform(n, mfact)
[A051142_row(n) for n in (0..9)] # Peter Luschny, Dec 31 2015

a(n, m) = a(n-1, m-1) - 4*(n-1)*a(n-1, m) for n >= m >= 1; a(n, m) := 0 for n < m; a(n, 0) := 0 for n >= 1; a(0, 0) = 1.
E.g.f. for the m-th column of the signed triangle: (log(1 + 4*x)/4)^m/m!.
a(n, m) = S1(n, m)*4^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

A002801 a(n) = (2n-1)a(n-1) - (n-1)*a(n-2) with a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 8, 50, 418, 4348, 54016, 779804, 12824540, 236648024, 4841363104, 108748223128, 2660609220952, 70422722065040, 2005010410792832, 61098981903602192, 1984186236246187024, 68407835576255308576, 2495374564069015050880, 96019859122742736121376, 3886906732751071879958816, 165120572466718493379680192

Offset: 0

N. J. A. Sloane

nonn

Row sums of A152148. - Paul Barry, Nov 26 2008

			G.f. = 1 + x + 2*x^2 + 8*x^3 + 50*x^4 + 418*x^5 + 4348*x^6 + 54016*x^7 + 779804*x^8 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..100
E. Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 223.
E. Lucas, Theorie des nombres (annotated scans of a few selected pages)
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
J. J. Sylvester, Note on determinants and duadic disynthemes, American J of Math, Vol 2 No 1, (1879), 89-96, circa p. 94.

Cf. A247249.

nxt[{n_,a_,b_}]:={n+1,b,b*(2n+1)-a*n}; Transpose[NestList[nxt,{1,1,1},30]][[2]] (* Harvey P. Dale, Sep 04 2013 *)
a[n_] := HypergeometricPFQ[{1/4, -n}, {}, -4]/(2^n); Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Mar 17 2014, after Mark van Hoeij *)
a[ n_] := If[ n < 0, 0,  n! SeriesCoefficient[ Exp[x/2] / (1 - 2 x)^(1/4), {x, 0, n}]]; (* Michael Somos, Oct 30 2015 *)
a[ n_] := If[ n < 0, 0, RecurrenceTable[{a[k] == (2 k - 1) a[k - 1] - (k - 1) a[k - 2], a[0] == a[1] == 1}, a, {k, n, n}]]; (* Michael Somos, Oct 30 2015 *)

a(n):=coeff(taylor(exp(x/2)/(1-2*x)^(1/4),x,0,n),x,n)*n!;
makelist(a(n),n,0,12); /* Emanuele Munarini, Jul 07 2011 */

x='x+O('x^66); /* that many terms */
Vec(serlaplace(exp(x/2)*(1-2*x)^(-1/4))) /* show terms */ /* Joerg Arndt, Jul 10 2011 */

Appears to be the BinomialMean transform of A007696 (see A075271). - John W. Layman, Oct 01 2002
E.g.f.: exp(x/2)*(1-2*x)^(-1/4). - Paul Barry, Nov 26 2008
a(n) = hypergeom([1/4, -n],[],-4)/(2^n). - Mark van Hoeij, Jun 02 2010
a(n) ~ n^(n-1/4) * exp(-n+1/4) * Gamma(3/4) * 2^n / sqrt(Pi). - Vaclav Kotesovec, Oct 08 2013
0 = a(n)*(+a(n+1) - 3*a(n+2) + a(n+3)) + a(n+1)*(-a(n+1) + 3*a(n+2) - 2*a(n+3)) + a(n+2)*(+2*a(n+2)) if n>=0. - Michael Somos, Oct 30 2015

More terms from John W. Layman, Oct 01 2002

A034908 One half of octo-factorial numbers.

Original entry on oeis.org

1, 10, 180, 4680, 159120, 6683040, 334152000, 19380816000, 1279133856000, 94655905344000, 7761784238208000, 698560581438720000, 68458936980994560000, 7256647319985423360000, 827257794478338263040000, 100925450926357268090880000, 13120308620426444851814400000

Offset: 1

Wolfdieter Lang

G. C. Greubel, Table of n, a(n) for n = 1..333
Index entries for sequences related to factorial numbers.

Cf. A007696, A045755, A034909, A039410, A039411, A039412, A084948.

[(&*[(8*k-6): k in [1..n]])/2: n in [1..30]]; // G. C. Greubel, Feb 26 2018

Drop[With[{nn = 40}, CoefficientList[Series[(-1 + (1 - 8*x)^(-1/4))/2, {x, 0, nn}], x]*Range[0, nn]!], 1] (* G. C. Greubel, Feb 26 2018 *)

my(x='x+O('x^30)); Vec(serlaplace((-1+(1-8*x)^(-1/4))/2)) \\ G. C. Greubel, Feb 26 2018

2*a(n) = (8*n-6)(!^8) := product(8*j-6, j=1..n) = 2^n*A007696(n); compare with A007696(n) = (4*n-3)(!^4) := product(4*j-3, j=1..n).
E.g.f.: (-1+(1-8*x)^(-1/4))/2.
G.f.: x/(1-10x/(1-8x/(1-18x/(1-16x/(1-26x/(1-24x/(1-34x/(1-32x/(1-... (continued fraction). - Philippe Deléham, Jan 07 2012
From Amiram Eldar, Dec 20 2022: (Start)
a(n) = A084948(n)/2.
Sum_{n>=1} 1/a(n) = 2*(e/8^6)^(1/8)*(Gamma(1/4) - Gamma(1/4, 1/8)). (End)

Terms a(16) onward added by G. C. Greubel, Feb 26 2018

A181048 Decimal expansion of (log(1+sqrt(2))+Pi/2)/(2sqrt(2)) = Sum_{k>=0} (-1)^k/(4k+1).

Original entry on oeis.org

8, 6, 6, 9, 7, 2, 9, 8, 7, 3, 3, 9, 9, 1, 1, 0, 3, 7, 5, 7, 3, 9, 9, 5, 1, 6, 3, 8, 8, 2, 8, 7, 0, 7, 1, 3, 6, 5, 2, 1, 7, 5, 3, 6, 7, 3, 4, 5, 2, 4, 4, 9, 0, 4, 3, 3, 5, 0, 3, 1, 8, 3, 8, 9, 1, 7, 6, 3, 9, 3, 5, 1, 4, 1, 0, 9, 4, 1, 3, 2, 9, 0, 5, 5, 7, 5, 0, 4, 0, 3, 4, 6, 3, 4, 0, 8, 9, 6, 8, 7, 0, 5, 2, 1, 8

Offset: 0

Jonathan D. B. Hodgson, Oct 01 2010, Oct 06 2010

			0.86697298733991103757399516388287071365217536734524490433....
At N = 100000 the truncated series 2*Sum_{k = 0..N/4 - 1} (-1)^k/(4*k + 1) = 1.7339(3)5974(5)7982(5)075(25)79(846)27(404)7... to 32 digits. The bracketed numbers show where this decimal expansion differs from that of 2*A181048. The numbers 1, 1, -3, -11, 57, 361 must be added to the bracketed numbers to give the correct decimal expansion to 32 digits: 2*( (log(1 + sqrt(2)) + Pi/2)/(2*sqrt(2)) ) = 1.7339(4)5974(6)7982(2)075(14)79(903)27(765)7.... - _Peter Bala_, Sep 23 2016

Jolley, Summation of Series, Dover (1961) eq 82 page 16.
Murray R. Spiegel, Seymour Lipschutz, John Liu. Mathematical Handbook of Formulas and Tables, 3rd Ed. Schaum's Outline Series. New York: McGraw-Hill (2009): p. 135, equation 21.17

Ivan Panchenko, Table of n, a(n) for n = 0..1000
J. M. Borwein, P. B. Borwein, and K. Dilcher, Pi, Euler numbers and asymptotic expansions, Amer. Math. Monthly, 96 (1989), 681-687.
Eric W. Weisstein, Euler's Series Transformation.
Herbert S. Wilf, Accelerated series for universal constants, by the WZ method, Discrete Mathematics & Theoretical Computer Science, Vol 3, No 4 (1999).

Cf. A001586, A003881, A024396, A091648, A093954, A113476, A181049, A181122, A188458, A193534, A262246.

RealDigits[(Pi Sqrt[2])/8 + (Sqrt[2] Log[1 + Sqrt[2]])/4, 10, 100][[1]] (* Alonso del Arte, Aug 11 2011 *)

(log(1+sqrt(2))+Pi/2)/(2*sqrt(2)) \\ G. C. Greubel, Jul 05 2017

(asinh(1)+Pi/2)/sqrt(8) \\ Charles R Greathouse IV, Jul 06 2017

Equals (A093954 + A091648/sqrt(2))/2.
Integral_{x = 0..1} 1/(1+x^4) = Sum_{k >= 0} (-1)^k/(4*k+1) = (log(1+sqrt(2)) + Pi/2)/(2*sqrt(2)).
1 - 1/5 + 1/9 - 1/13 + 1/17 - ... = (Pi*sqrt(2))/8 + (sqrt(2)*log(1 + sqrt(2)))/4 = (Pi + 2*log(1 + sqrt(2)))/(4 sqrt(2)). The first two are the formulas as given in Spiegel et al., the third is how Mathematica rewrites the infinite sum. - Alonso del Arte, Aug 11 2011
Let N be a positive integer divisible by 4. We have the asymptotic expansion 2*( (log(1 + sqrt(2)) + Pi/2)/(2*sqrt(2)) - Sum_{k = 0..N/4 - 1} (-1)^k/(4*k + 1) ) ~ 1/N + 1/N^2 - 3/N^3 - 11/N^4 + 57/N^5 + 361/N^6 - ..., where the sequence of coefficients [1, 1, -3, -11, 57, 361, ...] is A188458. This follows from Borwein et al., Lemma 2 with f(x) = 1/x and then set x = N/4 and h = 1/4. An example is given below. Cf. A181049. - Peter Bala, Sep 23 2016
Equals Sum_{n >= 0} 2^(n-1)*n!/(Product_{k = 0..n} 4*k + 1) = Sum_{n >= 0} 2^(n-1)*n!/A007696(n+1) (apply Euler's series transformation to Sum_{k >= 0} (-1)^k/(4*k + 1)). - Peter Bala, Dec 01 2021
From Peter Bala, Oct 23 2023: (Start)
The slowly converging series representation Sum_{n >= 0} (-1)^n/(4*n + 1) for the constant can be accelerated to give the following faster converging series:
1/2 + 2*Sum_{n >= 0} (-1)^n/((4*n + 1)(4*n + 5));
7/10 + 8*Sum_{n >= 0} (-1)^n/((4*n + 1)(4*n + 5)*(4*n + 9));
71/90 + 48*Sum_{n >= 0} (-1)^n/((4*n + 1)(4*n + 5)*(4*n + 9)*(4*n + 13));
971/1170 + 384*Sum_{n >= 0} (-1)^n/((4*n + 1)(4*n + 5)*(4*n + 9)*(4*n + 13)*(4*n + 17)).
These results may be easily verified by taking the partial fraction expansions of the summands. The general result appears to be that for r >= 0, the constant equals
C(r) + (2^r)*r!*Sum_{n >= 0} (-1)^n/((4*n + 1)*(4*n + 5)*...*(4*n + 4*r + 1)), where C(r) is the rational number Sum_{k = 0..r-1} 2^(k-1)*k!/(1*5*9*...*(4*k + 1)). [added 19 Feb 2024: the general result can be proved by the WZ method as described in Wilf.]
In the limit as r -> oo we find that the constant equals Sum_{k >= 0} 2^(k-1)*k!/(Product_{i = 0..k} 4*i + 1) as noted above. (End)
From Peter Bala, Mar 03 2024: (Start)
Continued fraction: 1/(1 + 1^2/(4 + 5^2/(4 + 9^2/(4 + 13^2/(4 + ... ))))) due to Euler.
Equals hypergeom([1/4, 1], [5/4], -1).
Gauss's continued fraction: 1/(1 + 1^2/(5 + 4^2/(9 + 5^2/(13 + 8^2/(17 + 9^2/(21 + 12^2/(25 + 13^2/(29 + 16^2/(33 + 17^2/(37 + ... )))))))))). (End)

cp's OEIS Frontend

A045755 8-fold factorials: a(n) = Product_{k=0..n-1} (8*k+1).

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A045756 Expansion of e.g.f. (1-9*x)^(-1/9), 9-factorial numbers.

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A034176 One third of quartic factorial numbers.

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A034177 a(n) is the n-th quartic factorial number divided by 4.

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A111146 Triangle T(n,k), read by rows, given by [0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938.

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A114799 Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.

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A002801 a(n) = (2n-1)a(n-1) - (n-1)*a(n-2) with a(0) = a(1) = 1.

A181048 Decimal expansion of (log(1+sqrt(2))+Pi/2)/(2sqrt(2)) = Sum_{k>=0} (-1)^k/(4k+1).