A047053 -id:A047053 - cp's OEIS frontend

A257612 Triangle read by rows: 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) = 4*x + 2.

1, 2, 2, 4, 24, 4, 8, 184, 184, 8, 16, 1216, 3680, 1216, 16, 32, 7584, 53824, 53824, 7584, 32, 64, 46208, 674752, 1507072, 674752, 46208, 64, 128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128, 256, 1677312, 84892672, 636233728, 1196803584, 636233728, 84892672, 1677312, 256

Offset: 0

Dale Gerdemann, May 06 2015

Corresponding entries in this triangle and in A060187 differ only by powers of 2. - F. Chapoton, Nov 04 2020

			Triangle begins as:
    1;
    2,      2;
    4,     24,       4;
    8,    184,     184,        8;
   16,   1216,    3680,     1216,       16;
   32,   7584,   53824,    53824,     7584,      32;
   64,  46208,  674752,  1507072,   674752,   46208,     64;
  128, 278912, 7764096, 33244544, 33244544, 7764096, 278912, 128;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A047053 (row sums), A060187, A142459, A257621.
Cf. A038208, A256890, A257609, A257610, A257614, A257616, A257617, A257618, A257619.
See similar sequences listed in A256890.

T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
Table[T[n,k,4,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

f(x) = 4*x + 2;
T(n, k) = t(n-k, k);
t(n, m) = if (!n && !m, 1, if (n < 0 || m < 0, 0, f(m)*t(n-1,m) + f(n)*t(n,m-1)));
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print();); \\ Michel Marcus, May 06 2015

def T(n,k,a,b): # A257612
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,4,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 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) = 4*x + 2.
Sum_{k=0..n} T(n,k) = A047053(n).
T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 4, and b = 2. - G. C. Greubel, Mar 20 2022

A051617 a(n) = (4n+5)(!^4)/5(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 9, 117, 1989, 41769, 1044225, 30282525, 999323325, 36974963025, 1515973484025, 68218806781125, 3342721532275125, 177164241210581625, 10098361749003152625, 616000066689192310125, 40040004334797500158125

Offset: 0

Wolfdieter Lang

Row m=5 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1) (rows m=0..4).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(9/4))); [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, 8, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(9/4), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(9/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+5)(!^4))/5(!^4).

E.g.f.: 1/(1-4*x)^(9/4).

A225471 Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

Original entry on oeis.org

1, 3, 1, 21, 10, 1, 231, 131, 21, 1, 3465, 2196, 446, 36, 1, 65835, 45189, 10670, 1130, 55, 1, 1514205, 1105182, 290599, 36660, 2395, 78, 1, 40883535, 31354119, 8951355, 1280419, 101325, 4501, 105, 1, 1267389585, 1012861224, 308846124, 48644344, 4421494, 240856, 7756, 136, 1

Offset: 0

Peter Luschny, May 17 2013

The Stirling-Frobenius cycle numbers are defined in A225470.

Triangle T(n,k), read by rows, given by (3, 4, 7, 8, 11, 12, 15, 16, 19, 20, ... (A014601)) DELTA (1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 14 2015

			[n\k][    0,       1,      2,     3,    4,  5,  6 ]
[0]       1,
[1]       3,       1,
[2]      21,      10,      1,
[3]     231,     131,     21,     1,
[4]    3465,    2196,    446,    36,    1,
[5]   65835,   45189,  10670,  1130,   55,  1,
[6] 1514205, 1105182, 290599, 36660, 2395, 78,  1.
...
From _Wolfdieter Lang_, Aug 11 2017: (Start)
Recurrence: T(4, 2) = T(3, 1) + (4*4 - 1)*T(3, 2) = 131 +15*21 = 446.
Boas-Buck recurrence for column k=2 and n=4: T(4, 2) = (4!/2)*(4*(3+8*(5/12)) *T(2, 2)/2! + 1*(3 + 8*(1/2))*T(3,2)/3!) = (4!/2)*(4*(19/3)/2  + 7*21/3!) =  446.
(End)

P. Bala, A 3 parameter family of generalized Stirling numbers.
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.

Columns k=0..3 give A008545, A286723(n-1), A383702, A383703.

Cf. A132393 (m=1), A028338 (m=2), A225470 (m=3).

T[0, 0] = 1; T[n_, k_] := Sum[Binomial[n - j, k]*Abs[StirlingS1[n, n - j]]* 3^(n - k - j)*4^j, {j, 0, n - k}];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 30 2018, after Wolfdieter Lang *)

@CachedFunction
def SF_C(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return SF_C(n-1, k-1, m) + (m*n-1)*SF_C(n-1, k, m)
for n in (0..8): [SF_C(n, k, 4) for k in (0..n)]

For a recurrence see the Sage program.
T(n, 0) ~ A008545; T(n, n) ~ A000012; T(n, n-1) = A014105.
Row sums ~ A047053; alternating row sums ~ A001813.
From Wolfdieter Lang, May 29 2017: (Start)
This is the Sheffer triangle (1/(1 - 4*x)^{-3/4}, -(1/4)*log(1-4*x)). See the P. Bala link where this is called exponential Riordan array, and the signed version is denoted by s_{(4,0,3)}.
E.g.f. of row polynomials in the variable x (i.e., of the triangle): (1 - 4*z)^{-(3+x)/4}.
E.g.f. of column k: (1-4*x)^(-3/4)*(-(1/4)*log(1-4*x))^k/k!, k >= 0.
Recurrence for row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k: R(n, x) = (x+3)*R(n-1,x+4), with R(0, x) = 1.
R(n, x) = risefac(4,3;x,n) := Product_{j=0..(n-1)} (x + (3 + 4*j)). (See the P. Bala link, eq. (16) for the signed s_{4,0,3} row polynomials.)
T(n, k) = Sum_{j=0..(n-m)} binomial(n-j, k)* S1p(n, n-j)*3^(n-k-j)*4^j, with S1p(n, m) = A132393(n, m).
T(n, k) = sigma[4,3]^{(n)}_{n-k}, with the elementary symmetric functions sigma[4,3]^{(n)}_m of degree m in the n numbers 3, 7, 11, ..., 3+4*(n-1), with sigma[4,3]^{(n)}_0 := 1. (End)
Boas-Buck type recurrence for column sequence k: T(n, k) = (n!/(n - k)) * Sum_{p=k..n-1} 4^(n-1-p)*(3 + 8*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, and beta(k) = A002208(k+1)/A002209(k+1), beginning with {1/2, 5/12, 3/8, 251/720, ...}. See a comment and references in A286718. - Wolfdieter Lang, Aug 11 2017

A051618 a(n) = (4*n+6)(!^4)/6(!^4).

Original entry on oeis.org

1, 10, 140, 2520, 55440, 1441440, 43243200, 1470268800, 55870214400, 2346549004800, 107941254220800, 5397062711040000, 291441386396160000, 16903600410977280000, 1048023225480591360000, 69169532881719029760000, 4841867301720332083200000, 358298180327304574156800000

Offset: 0

Wolfdieter Lang

This sequence is related to A000407 ((4*n+2)(!^4) quartic, or 4-factorials).
Row m=6 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.
a(n) = A001813 a(n+2)/12. - Zerinvary Lajos, Feb 15 2008
For n>4, a(n) mod n^2 = n*(n-2) if n is prime, otherwise 0. - Gary Detlefs, Apr 16 2012

G. C. Greubel, Table of n, a(n) for n = 0..360

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617 through A051622 (rows m=0..10).

[Factorial(2*n+4)/(12*Factorial(n+2)): n in [0..100]]; // Vincenzo Librandi, Jul 04 2015

seq(mul((n+2+k), k=1..n+2)/12, n=0..17); # Zerinvary Lajos, Feb 15 2008
A051618 := n -> 2^n*(n+1)!*JacobiP(n+1, 1/2, -(n+1), 3)/3:
seq(simplify(A051618(n)), n = 0..19);  # Peter Luschny, Jan 22 2025

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 9, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
f[n_] := (2n + 4)!/(12(n + 2)!); Array[f, 16, 0] (* Or *)
FoldList[ #2*#1 &, 1, Range[10, 66, 4]] (* Robert G. Wilson v *)
With[{nn=20},CoefficientList[Series[1/(1-4x)^(5/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 24 2015 *)
Table[(Product[(4*k + 6), {k, 0, n}])/6, {n, 0, 50}] (* G. C. Greubel, Jan 27 2017 *)

A051618(n):=(2*n+4)!/(12*(n+2)!)$
makelist(A051618(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

for(n=0,25, print1((2*n+3)!/(6*(n+1)!), ", ")) \\ G. C. Greubel, Jan 27 2017

a(n) = ((4*n+6)(!^4))/6(!^4).
E.g.f.: 1/(1-4*x)^(5/2).
a(n) = (2n+4)!/(12(n+2)!). -  Gary Detlefs, Mar 06 2011
a(n) = (2*n+3)!/(6*(n+1)!). - Gary Detlefs, Apr 16 2012
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+5)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
a(n) = (4^(1+n)*Gamma(5/2+n))/(3*sqrt(Pi)). - Gerry Martens, Jul 02 2015
a(n) ~ 2^(2*n+5/2) * n^(n+2) / (3*exp(n)). - Vaclav Kotesovec, Jul 04 2015
a(n) = 2^n*(n+1)!*JacobiP(n+1, 1/2, -(n+1), 3)/3. - Peter Luschny, Jan 22 2025

A051622 a(n) = (4n+10)(!^4)/10(!^4), related to A000407 ((4n+2)(!^4) quartic, or 4-factorials).

Original entry on oeis.org

1, 14, 252, 5544, 144144, 4324320, 147026880, 5587021440, 234654900480, 10794125422080, 539706271104000, 29144138639616000, 1690360041097728000, 104802322548059136000, 6916953288171902976000, 484186730172033208320000

Offset: 0

Wolfdieter Lang

Row m=10 of the array A(5; m,n) := ((4*n+m)(!^4))/m(!^4), m >= 0, n >= 0.
From Zerinvary Lajos, Feb 15 2008: (Start)
a(n) = A001813(n+3)/120.
a(n) = A051618(n+1)/10. (End)

G. C. Greubel, Table of n, a(n) for n = 0..363

Cf. A047053, A007696(n+1), A000407, A034176(n+1), A034177(n+1), A051617-A051621 (rows m=0..9).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-4*x)^(14/4))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

seq(mul((n+3+k), k=1..n+3)/120, n=0..18); # Zerinvary Lajos, Feb 15 2008

s=1;lst={s};Do[s+=n*s;AppendTo[lst, s], {n, 13, 5!, 4}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 4*x)^(7/2), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-4*x)^(14/4))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((4*n+10)(!^4))/10(!^4) = A000407(n+2)/(6*10).
E.g.f.: 1/(1-4*x)^(7/2).
G.f.: G(0)/2, where G(k)= 1 + 1/(1 - 2*x/(2*x + 1/(2*k+7)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013

A225118 Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 4}(x) in descending order.

Original entry on oeis.org

1, 3, 1, 9, 22, 1, 27, 235, 121, 1, 81, 1996, 3446, 620, 1, 243, 15349, 63854, 40314, 3119, 1, 729, 112546, 963327, 1434812, 422087, 15618, 1, 2187, 806047, 12960063, 37898739, 26672209, 4157997, 78117, 1, 6561, 5705752, 162711868, 840642408, 1151050534

Offset: 0

Peter Luschny, May 02 2013

The row sums equal the quadruple factorial numbers A047053 and the alternating row sums, i.e., sum((-1)^k*T(n,k),k=0..n), are up to a sign A079858. - Johannes W. Meijer, May 04 2013

			[0]  1
[1]  3*x   +    1
[2]  9*x^2 +   22*x   +    1
[3] 27*x^3 +  235*x^2 +  121*x   + 1
[4] 81*x^4 + 1996*x^3 + 3446*x^2 + 620*x + 1
...
The triangle T(n, k) begins:
n\k
0:    1
1:    3      1
2:    9     22        1
3:   27    235      121        1
4:   81   1996     3446      620        1
5:  243  15349    63854    40314     3119       1
6:  729 112546   963327  1434812   422087   15618     1
7: 2187 806047 12960063 37898739 26672209 4157997 78117 1
...
row n=8: 6561 5705752 162711868 840642408 1151050534 442372648 39531132 390616 1,
row n=9: 19683 40156777 1955297356 16677432820 39523450714 29742429982 6818184988 367889284 1953115 1.
... - _Wolfdieter Lang_, Apr 12 2017

Peter Luschny, Generalized Eulerian polynomials.
Zhe Wang and Zhi-Yong Zhu, The spiral property of q-Eulerian numbers of type B, The Australasian Journal of Combinatorics, Volume 87(1) (2023), Pages 198-202. See p. 199.

Coefficients of A_{n,1}(x) = A008292, coefficients of A_{n,2}(x) = A060187, coefficients of A_{n,3}(x) = A225117. A123125, A225467, A225469, A225473.

gf := proc(n, k) local f; f := (x,t) -> x*exp(t*x/k)/(1-x*exp(t*x));
series(f(x,t), t, n+2); ((1-x)/x)^(n+1)*k^n*n!*coeff(%, t, n):
collect(simplify(%), x) end:
seq(print(seq(coeff(gf(n, 4), x, n-k), k=0..n)), n=0..6);
# Recurrence:
P := proc(n,x) option remember; if n = 0 then 1 else
  (n*x+(1/4)*(1-x))*P(n-1,x)+x*(1-x)*diff(P(n-1,x),x);
  expand(%) fi end:
A225117 := (n,k) -> 4^n*coeff(P(n,x),x,n-k):
seq(print(seq(A225117(n,k), k=0..n)), n=0..5);  # Peter Luschny, Mar 08 2014

gf[n_, k_] := Module[{f, s}, f[x_, t_] := x*Exp[t*x/k]/(1-x*Exp[t*x]); s = Series[f[x, t], {t, 0, n+2}]; ((1-x)/x)^(n+1)*k^n*n!*SeriesCoefficient[s, {t, 0, n}]]; Table[Table[SeriesCoefficient[gf[n, 4], {x, 0, n-k}], {k, 0, n}], {n, 0, 8}] // Flatten (* Jean-François Alcover, Jan 27 2014, after Maple *)

@CachedFunction
def EB(n, k, x):  # Modified cardinal B-splines
    if n == 1: return 0 if (x < 0) or (x >= 1) else 1
    return k*x*EB(n-1, k, x) + k*(n-x)*EB(n-1, k, x-1)
def EulerianPolynomial(n, k): # Generalized Eulerian polynomials
    R. = ZZ[]
    if x == 0: return 1
    return add(EB(n+1, k, m+1/k)*x^m for m in (0..n))
[EulerianPolynomial(n, 4).coefficients()[::-1] for n in (0..5)]

G.f. of the polynomials is gf(n, k) = k^n*n!*(1/x-1)^(n+1)[t^n](x*e^(t*x/k)*(1-x*e(t*x))^(-1)) for k = 4; here [t^n]f(t,x) is the coefficient of t^n in f(t,x).
From Wolfdieter Lang, Apr 12 2017 : (Start)
E.g.f. of row polynomials (rising powers of x): (1-x)*exp(3*(1-x)*z)/(1-y*exp(4*(1-x)*z)), i.e. e.g.f. of the triangle.
E.g.f. for the row polynomials with falling powers of x (A_{n, 4}(x) of the name): (1-x)*exp((1-x)*z)/(1 - x*exp(4*(1-x)*z)).
T(n, k) = Sum_{j=0..k} (-1)^(k-j) * binomial(n+1,k-j) * (3+4*j)^n, 0 <= k <= n.
Recurrence: T(n, k) = (4*(n-k) + 1)*T(n-1, k-1) + (3 + 4*k)*T(n-1, k), n >= 1, with T(n, -1) = 0, T(0, 0) = 1 and T(n, k) = 0 for n < k. (End)
In terms of Euler's triangle = A123125: T(n, k) = Sum_{m=0..n} (binomial(n, m)*3^(n-m)*4^m*Sum_{p=0..k} (-1)^(k-p)*binomial(n-m, k-p)*A123125(m, p)), 0 <= k <= n. - Wolfdieter Lang, Apr 13 2017

A303487 a(n) = n! * [x^n] 1/(1 - 4*x)^(n/4).

Original entry on oeis.org

1, 1, 12, 231, 6144, 208845, 8648640, 422463195, 23781703680, 1515973484025, 107941254220800, 8491022274509775, 731304510986649600, 68444451854354701125, 6916953288171902976000, 750681472158682148959875, 87076954662428278259712000, 10751175443940144673035200625

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

			a(1) = 1;
a(2) = 2*6 = 12;
a(3) = 3*7*11 = 231;
a(4) = 4*8*12*16 = 6144;
a(5) = 5*9*13*17*21 = 208845, etc.

Index entries for sequences related to factorial numbers

Column k=4 of A303489.

Cf. A000407, A001813, A007696, A008545, A034176, A034177, A047053, A051617, A051618, A051619, A051620, A051621, A051622, A113551, A303486, A303488.

Table[n! SeriesCoefficient[1/(1 - 4 x)^(n/4), {x, 0, n}], {n, 0, 17}]
Table[Product[4 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[4^n Pochhammer[n/4, n], {n, 0, 17}]

a(n) = Product_{k=0..n-1} (4*k + n).
a(n) = 4^n*Gamma(5*n/4)/Gamma(n/4).
a(n) ~ 5^(5*n/4-1/2)*n^n/exp(n).

A087299 Ratio of volume of n-dimensional ball to circumscribing n-cube is Pi^floor(n/2) divided by a(n).

Original entry on oeis.org

1, 1, 4, 6, 32, 60, 384, 840, 6144, 15120, 122880, 332640, 2949120, 8648640, 82575360, 259459200, 2642411520, 8821612800, 95126814720, 335221286400, 3805072588800, 14079294028800, 167423193907200, 647647525324800

Offset: 0

Eric W. Weisstein, Aug 31 2003

			The volume of sphere (3-ball) is 4/3*Pi*r^3 and circumscribing 3-cube is 2^3*r^3 so ratio is Pi/6 and a(3)=6.
G.f. =  1 + x + 4*x^2 + 6*x^3 + 32*x^4 + 60*x^5 + 384*x^6 + 840*x^7 + ...

N. Cakic, D. Letic, B. Davidovic, The Hyperspherical functions of a derivative, Abstr. Appl. Anal. (2010) 364292 doi:10.1155/2010/364292

G. C. Greubel, Table of n, a(n) for n = 0..725
Dusko Letic, Nenad Cakic, Branko Davidovic and Ivana Berkovic, Orthogonal and diagonal dimension fluxes of hyperspherical function, Advances in Difference Equations 2012, 2012:22. - From _N. J. A. Sloane_, Sep 04 2012
Eric Weisstein's World of Mathematics, Ball

Cf. A000407, A047053, A072345, A072346.

a[ n_] := If[ n < 0, 0, With[ {m = n + 1}, m! SeriesCoefficient[ Exp[x^2] (1 + Sqrt[Pi] Erf[x]), {x, 0, m}] / 2]]; (* Michael Somos, Jan 24 2014 *)
Table[2^n*Gamma[n/2 + 1]*Pi^Floor[n/2]/Pi^(n/2), {n,0,50}] (* G. C. Greubel, Jan 28 2017 *)

{a(n) = my(A); if( n<0, 0, n++; A = exp(x^2 + x * O(x^n)); n! * polcoeff( A * (1 + 2 * intformal( 1/A )), n) / 2)}; /* Michael Somos, May 25 2004 */

{a(n) = if( n<2, n>-1, 2*n * a(n-2))}; /* Michael Somos, Jan 24 2014 */

{a(n) = if( n<0, 0, if( n%2, n! / (n\2)!, 2^n * (n\2)!))}; /* Michael Somos, Jan 03 2015 */

a(n) = 2^n*gamma(n/2+1)*Pi^floor(n/2)/Pi^(n/2), n >= 0. - Wolfdieter Lang, Jul 17 2013
0 = a(n)*( 2*a(n+1) - a(n+3) ) + a(n+1)*a(n+2) if n>=0. - Michael Somos, Jan 24 2014
a(n) = 2*n * a(n-2) if n>=2. - Michael Somos, Jan 24 2014
a(2*n) = A047053(n). a(2*n + 1) = A000407(n). - Michael Somos, Jan 03 2015

A144829 Partial products of successive terms of A017209; a(0)=1 .

Original entry on oeis.org

1, 4, 52, 1144, 35464, 1418560, 69509440, 4031547520, 270113683840, 20528639971840, 1744934397606400, 164023833375001600, 16894454837625164800, 1892178941814018457600, 228953651959496233369600, 29763974754734510338048000, 4137192490908096936988672000

Offset: 0

Philippe Deléham, Sep 21 2008

			a(0)=1, a(1)=4, a(2)=4*13=52, a(3)=4*13*22=1144, a(4)=4*13*22*31=35464, ...

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A001715, A002866, A007559, A008546, A047053.

Cf. A048994, A049308, A132393, A144827, A144828.

[n le 2 select 4^(n-1) else (9*n-14)*Self(n-1): n in [1..30]]; // G. C. Greubel, May 26 2022

Table[4*9^(n-1)*Pochhammer[13/9, n-1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 29 2021 *)

a(n) = (-5)^n*sum(k=0, n, (9/5)^k*stirling(n+1,n+1-k, 1)); \\ Michel Marcus, Feb 20 2015

[9^n*rising_factorial(4/9, n) for n in (0..30)] # G. C. Greubel, May 26 2022

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*9^(n-k).
a(n) = (-5)^n*Sum_{k=0..n} (9/5)^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
a(n) + (5-9*n)*a(n-1) = 0. - R. J. Mathar, Sep 04 2016
From Vaclav Kotesovec, Nov 29 2021: (Start)
a(n) = 9^n * Gamma(n + 4/9) / Gamma(4/9).
a(n) ~ sqrt(2*Pi) * 9^n * n^(n - 1/18) / (Gamma(4/9) * exp(n)). (End)
From G. C. Greubel, May 26 2022: (Start)
G.f.: hypergeometric2F0([1, 4/9], [], 9*x).
E.g.f.: (1-9*x)^(-4/9). (End)
Sum_{n>=0} 1/a(n) = 1 + (e/9^5)^(1/9)*(Gamma(4/9) - Gamma(4/9, 1/9)). - Amiram Eldar, Dec 21 2022

a(9) originally given incorrectly as 20520639971840 corrected by Peter Bala, Feb 20 2015

A167569 The lower left triangle of the ED2 array A167560.

Original entry on oeis.org

1, 2, 4, 6, 16, 32, 24, 80, 192, 384, 120, 480, 1344, 3072, 6144, 720, 3360, 10752, 27648, 61440, 122880, 5040, 26880, 96768, 276480, 675840, 1474560, 2949120, 40320, 241920, 967680, 3041280, 8110080, 19169280, 41287680, 82575360

Offset: 1

Johannes W. Meijer, Nov 10 2009

We discovered that the numbers that appear in the lower left triangle of the ED2 array A167560 (m <= n) behave in a regular way, see the formula below. This rather simple regularity doesn't show up in the upper right triangle of the ED2 array (m > n).

			The first few triangle rows are:
[1]
[2, 4]
[6, 16, 32]
[24, 80, 192, 384]
[120, 480, 1344, 3072, 6144]
[720, 3360, 10752, 27648, 61440, 122880]

G. C. Greubel, Table of n, a(n) for the first 50 rows

A167560 is the ED2 array.
A047053, 2*A034177 and A167570 are the first three right hand triangle columns.
A000142, 4*A001715, 32*A001725, 384* A049388 and 6144* A049398 are the first five left hand triangle columns.
A167571 equals the row sums.

a := proc(n, m): 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)! end: seq(seq(a(n, m), m=1..n), n=1..8); # Johannes W. Meijer, revised Nov 23 2012

Flatten[Table[4^(m - 1)*(m - 1)!*(n + m - 1)!/(2*m - 1)!, {n, 1, 50}, {m, n}]] (* G. C. Greubel, Jun 16 2016 *)

a(n,m) = 4^(m-1)*(m-1)!*(n+m-1)!/(2*m-1)!.

cp's OEIS Frontend

A257612 Triangle read by rows: 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) = 4*x + 2.

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A051617 a(n) = (4*n+5)(!^4)/5(!^4), related to A007696(n+1) ((4*n+1)(!^4) quartic, or 4-factorials).

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A225471 Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

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A051618 a(n) = (4*n+6)(!^4)/6(!^4).

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A051622 a(n) = (4*n+10)(!^4)/10(!^4), related to A000407 ((4*n+2)(!^4) quartic, or 4-factorials).

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A225118 Triangle read by rows, coefficients of the generalized Eulerian polynomials A_{n, 4}(x) in descending order.

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A303487 a(n) = n! * [x^n] 1/(1 - 4*x)^(n/4).

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A257612 Triangle read by rows: 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) = 4*x + 2.

A051617 a(n) = (4n+5)(!^4)/5(!^4), related to A007696(n+1) ((4n+1)(!^4) quartic, or 4-factorials).

A051622 a(n) = (4n+10)(!^4)/10(!^4), related to A000407 ((4n+2)(!^4) quartic, or 4-factorials).