A034001 -id:A034001 - cp's OEIS frontend

A034000 One half of triple factorial numbers.

1, 5, 40, 440, 6160, 104720, 2094400, 48171200, 1252451200, 36321084800, 1162274713600, 40679614976000, 1545825369088000, 63378840132608000, 2788668965834752000, 131067441394233344000, 6553372069711667200000, 347328719694718361600000, 19450408302904228249600000

Offset: 1

Wolfdieter Lang

Preface the series with a 1, then the next term = (1, 4, 7, 10, ...) dot (1, 1, 5, 40, ...). E.g., a(5) = 6160 = (1, 4, 7, 10, 13) dot (1, 1, 5, 40, 440) = (1 + 4 + 35 + 400 + 5720). - Gary W. Adamson, May 17 2010

In other words, a(n) = Sum_{i=0..n-1} b(i)*A016777(i) where b(0)=1 and b(n)=a(n). - Michel Marcus, Dec 18 2022

G. C. Greubel, Table of n, a(n) for n = 1..375
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.

Cf. A007559, A034001, A025748, A034724, A016777.

a:=[1];; for n in [2..20] do a[n]:=(3*n-1)*a[n-1]; od; a; # G. C. Greubel, Aug 15 2019

[n le 1 select 1 else (3*n-1)*Self(n-1): n in [1..20]]; // G. C. Greubel, Aug 15 2019

A034000:=n->`if`(n=1, 1, (3*n-1)*A034000(n-1)); seq(A034000(n), n=1..20); # G. C. Greubel, Aug 15 2019

nxt[{n_,a_}]:={n+1,(3(n+1)-1)*a}; Transpose[NestList[nxt,{1,1},20]][[2]] (* Harvey P. Dale, Aug 22 2015 *)
Table[3^(n-1)*Pochhammer[5/3, n-1], {n,20}] (* G. C. Greubel, Aug 15 2019 *)

m=20; v=concat([1], vector(m-1)); for(n=2, m, v[n]=(3*n-1)*v[n-1]); v \\ G. C. Greubel, Aug 15 2019

def a(n):
    if n==1: return 1
    else: return (3*n-1)*a(n-1)
[a(n) for n in (1..20)] # G. C. Greubel, Aug 15 2019

a(n) = A007661(3n-1)/2 = A008544(n)/2.
2*a(n+1) = (3*n+2)!!! = Product_{j=0..n} (3*j+2), n >= 0.
E.g.f.: (-1 + (1-3*x)^(-2/3))/2.
a(n) = (3*n-1)!/(2*3^(n-1)*(n-1)!*A007559(n)).
a(n) ~ 3/2*2^(1/2)*Pi^(1/2)*Gamma(2/3)^-1*n^(7/6)*3^n*e^-n*n^n*{1 + 23/36*n^-1 + ...}. - Joe Keane (jgk(AT)jgk.org), Nov 23 2001
a(n) = 3^n*(n+2/3)!/(2/3)!, with offset 0. - Paul Barry, Sep 04 2005
D-finite with recurrence a(n) + (1-3*n)*a(n-1) = 0. - R. J. Mathar, Dec 03 2012
Sum_{n>=1} 1/a(n) = 2*(e/3)^(1/3)*(Gamma(2/3) - Gamma(2/3, 1/3)). - Amiram Eldar, Dec 18 2022

A257620 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 3*n + 3.

Original entry on oeis.org

1, 3, 3, 9, 36, 9, 27, 297, 297, 27, 81, 2106, 5346, 2106, 81, 243, 13851, 73386, 73386, 13851, 243, 729, 87480, 868239, 1761264, 868239, 87480, 729, 2187, 540189, 9388791, 34158753, 34158753, 9388791, 540189, 2187, 6561, 3293622, 95843088, 578903274, 1024762590, 578903274, 95843088, 3293622, 6561

Offset: 0

Dale Gerdemann, May 09 2015

			Array t(n,k) begins as:
    1,      3,        9,         27,           81,            243, ...;
    3,     36,      297,       2106,        13851,          87480, ...;
    9,    297,     5346,      73386,       868239,        9388791, ...;
   27,   2106,    73386,    1761264,     34158753,      578903274, ...;
   81,  13851,   868239,   34158753,   1024762590,    25791697782, ...;
  243,  87480,  9388791,  578903274,  25791697782,   928501120152, ...;
  729, 540189, 95843088, 8959544136, 575025893586, 28788563928042, ...;
Triangle T(n,k) begins as:
     1;
     3,      3;
     9,     36,       9;
    27,    297,     297,       27;
    81,   2106,    5346,     2106,       81;
   243,  13851,   73386,    73386,    13851,     243;
   729,  87480,  868239,  1761264,   868239,   87480,    729;
  2187, 540189, 9388791, 34158753, 34158753, 9388791, 540189, 2187;

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

Cf. A000244, A008292, A034001 (row sums), A038221, A142458, A257180, A257610.
Cf. A257611, A257621, A257622, A257623, A257624, A257625, A257626, A257627.
Similar sequences listed in A256890.

A257620:= func< n,k | 3^n*EulerianNumber(n+1, k) >;
[A257620(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 17 2025

t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
T[n_, k_, p_, q_]= t[n-k, k, p, q];
Table[T[n,k,3,3], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 28 2022 *)

from sage.all import *
from sage.combinat.combinat import eulerian_number
def A257620(n,k): return pow(3,n)*eulerian_number(n+1,k)
print(flatten([[A257620(n,k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Jan 17 2025

@CachedFunction
def t(n,k,p,q):
    if (n<0 or k<0): return 0
    elif (n==0 and k==0): return 1
    else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
def A257620(n,k): return t(n-k,k,3,3)
flatten([[A257620(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 28 2022

T(n, k) = t(n-k, k), where 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), and f(n) = 3*n + 3.
Sum_{k=0..n} T(n, k) = A034001(n).
From G. C. Greubel, Feb 28 2022: (Start)
t(k, n) = t(n, k).
T(n, n-k) = T(n, k).
t(0, n) = T(n, 0) = A000244(n). (End)
T(n, k) = 3^n*A008292(n, k). - G. C. Greubel, Jan 17 2025

A051604 a(n) = (3*n+4)!!!/4!!!.

Original entry on oeis.org

1, 7, 70, 910, 14560, 276640, 6086080, 152152000, 4260256000, 132067936000, 4490309824000, 166141463488000, 6645658539520000, 285763317199360000, 13145112591170560000, 644110516967357440000, 33493746882302586880000, 1842156078526642278400000

Offset: 0

Wolfdieter Lang

Related to A007559(n+1) ((3*n+1)!!! triple factorials).

Row m=4 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..378

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1). (rows m=0..3).

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

With[{nn = 30}, CoefficientList[Series[1/(1-3*x)^(7/3), {x, 0, nn}], x]* Range[0,nn]! ] (* G. C. Greubel, Aug 15 2018 *)
With[{c=Times@@Range[4,1,-3]},Table[(Times@@Range[3n+4,1,-3])/c,{n,0,20}]] (* Harvey P. Dale, Feb 06 2023 *)

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

a(n) = ((3*n+4)(!^3))/4(!^3).
E.g.f.: 1/(1-3*x)^(7/3).
Sum_{n>=0} 1/a(n) = 1 + 3*(3*e)^(1/3)*(Gamma(7/3) - Gamma(7/3, 1/3)). - Amiram Eldar, Dec 23 2022

A051605 a(n) = (3*n+5)!!!/5!!!.

Original entry on oeis.org

1, 8, 88, 1232, 20944, 418880, 9634240, 250490240, 7264216960, 232454942720, 8135922995200, 309165073817600, 12675768026521600, 557733793166950400, 26213488278846668800, 1310674413942333440000, 69465743938943672320000, 3890081660580845649920000

Offset: 0

Wolfdieter Lang

Related to A008544(n+1) ((3*n+2)!!! triple factorials).

Row m=5 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..378

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604 (rows m=0..4).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(8/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

RecurrenceTable[{a[0]==1,a[n]==(3n+5)a[n-1]},a,{n,20}] (* Harvey P. Dale, Oct 19 2013 *)
With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(8/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(8/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((3*n+5)(!^3))/5(!^3).
E.g.f.: 1/(1-3*x)^(8/3).
a(n) = 3^n*(n+5/3)!/(5/3)!. - Paul Barry, Sep 04 2005
a(n) = (3*n+5)*a(n-1). - R. J. Mathar, Nov 13 2012
Sum_{n>=0} 1/a(n) = 1 + 3*(9*e)^(1/3)*(Gamma(8/3) - Gamma(8/3, 1/3)). - Amiram Eldar, Dec 23 2022

A051606 a(n) = (3n+6)!!!/6!!!, related to A032031 ((3n)!!! triple factorials).

Original entry on oeis.org

1, 9, 108, 1620, 29160, 612360, 14696640, 396809280, 11904278400, 392841187200, 14142282739200, 551549026828800, 23165059126809600, 1042427660706432000, 50036527713908736000, 2551862913409345536000, 137800597324104658944000, 7854634047473965559808000

Offset: 0

Wolfdieter Lang

Row m=6 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..378

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604-A051609 (rows m=0..9).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(9/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

[seq(n!*3^(n-2)/2, n=2..18)]; # Zerinvary Lajos, Sep 23 2006
with(combstruct):ZL:=[T,{T=Union(Z,Prod(Epsilon,Z,T),Prod(T,Z,Epsilon),Prod(T,Z))},labeled]:seq(count(ZL,size=i)/6,i=2..18); # Zerinvary Lajos, Dec 16 2007
restart: G(x):=(1-3*x)^(n-4): 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..16); # Zerinvary Lajos, Apr 04 2009

With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(9/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(9/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((3*n+6)(!^3))/6(!^3); e.g.f.: 1/(1-3*x)^3.
a(n) = n!*3^(n-2)/2, n >= 2. - Zerinvary Lajos, Sep 23 2006
Sum_{n>=0} 1/a(n) = 18*exp(1/3) - 24. - Amiram Eldar, Dec 18 2022

A051609 a(n) = (3n+9)!!!/9!!!, related to A032031 ((3n)!!! triple factorials).

Original entry on oeis.org

1, 12, 180, 3240, 68040, 1632960, 44089920, 1322697600, 43649020800, 1571364748800, 61283225203200, 2573895458534400, 115825295634048000, 5559614190434304000, 283540323712149504000, 15311177480456073216000, 872737116385996173312000, 52364226983159770398720000

Offset: 0

Wolfdieter Lang

Row m=9 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..377

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604-A051608 (rows m=0..8).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(12/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

restart: G(x):=(1-3*x)^(n-5): 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 04 2009

With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(12/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(12/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((3*n+9)(!^3))/9(!^3).
E.g.f.: 1/(1-3*x)^4.
From Amiram Eldar, Dec 18 2022: (Start)
a(n) = (n+3)!*3^(n-1)/2.
Sum_{n>=0} 1/a(n) = 162*exp(1/3) - 225. (End)

A303486 a(n) = n! * [x^n] 1/(1 - 3*x)^(n/3).

Original entry on oeis.org

1, 1, 10, 162, 3640, 104720, 3674160, 152152000, 7264216960, 392841187200, 23734494784000, 1584471003315200, 115825295634048000, 9201578813819392000, 789383453851632640000, 72728093032166347776000, 7162140885524461957120000, 750766815289210771251200000

Offset: 0

Ilya Gutkovskiy, Apr 24 2018

nonn

			a(1) = 1;
a(2) = 2*5 = 10;
a(3) = 3*6*9 = 162;
a(4) = 4*7*10*13 = 3640;
a(5) = 5*8*11*14*17 = 104720, etc.

G. C. Greubel, Table of n, a(n) for n = 0..343
Index entries for sequences related to factorial numbers

Column k=3 of A303489.

Cf. A000407, A007559, A008544, A032031, A034000, A034001, A051604, A051605, A051606, A051607, A051608, A051609, A113551, A303487, A303488.

Table[n! SeriesCoefficient[1/(1 - 3 x)^(n/3), {x, 0, n}], {n, 0, 17}]
Table[Product[3 k + n, {k, 0, n - 1}], {n, 0, 17}]
Table[3^n Pochhammer[n/3, n], {n, 0, 17}]

a(n) = Product_{k=0..n-1} (3*k + n).
a(n) = 3^n*Gamma(4*n/3)/Gamma(n/3).
a(n) ~ 2^(8*n/3-1)*n^n/exp(n).

A051607 a(n) = (3*n+7)!!!/7!!!.

Original entry on oeis.org

1, 10, 130, 2080, 39520, 869440, 21736000, 608608000, 18866848000, 641472832000, 23734494784000, 949379791360000, 40823331028480000, 1877873227310080000, 92015788138193920000, 4784820983186083840000, 263165154075234611200000, 15263578936363607449600000

Offset: 0

Wolfdieter Lang

Related to A007559(n+1) ((3*n+1)!!! triple factorials).

Row m=7 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..378

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604, A051605, A051606, A051608, A051609 (rows m=0..9).

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(10/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(10/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(10/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((3*n+7)(!^3))/7(!^3).
E.g.f.: 1/(1-3*x)^(10/3).
Sum_{n>=0} 1/a(n) = 1 + 9*(3*e)^(1/3)*(Gamma(10/3) - Gamma(10/3, 1/3)). - Amiram Eldar, Dec 23 2022

A051608 a(n) = (3*n+8)!!!/8!!!.

Original entry on oeis.org

1, 11, 154, 2618, 52360, 1204280, 31311280, 908027120, 29056867840, 1016990374400, 38645634227200, 1584471003315200, 69716724145868800, 3276686034855833600, 163834301742791680000, 8683217992367959040000, 486260207572605706240000, 28689352246783736668160000

Offset: 0

Wolfdieter Lang

Related to A008544(n+1) ((3*n+2)!!! triple factorials).

Row m=8 of the array A(4; m,n) := ((3*n+m)(!^3))/m(!^3), m >= 0, n >= 0.

G. C. Greubel, Table of n, a(n) for n = 0..377

Cf. A032031, A007559(n+1), A034000(n+1), A034001(n+1), A051604, A051605, A051606, A051607, A051609 (rows m=0..9).

Cf. A008544.

m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(1/(1-3*x)^(11/3))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 15 2018

With[{nn = 30}, CoefficientList[Series[1/(1 - 3*x)^(11/3), {x, 0, nn}], x]*Range[0, nn]!] (* G. C. Greubel, Aug 15 2018 *)

x='x+O('x^30); Vec(serlaplace(1/(1-3*x)^(11/3))) \\ G. C. Greubel, Aug 15 2018

a(n) = ((3*n+8)(!^3))/8(!^3).
E.g.f.: 1/(1-3*x)^(11/3).
Sum_{n>=0} 1/a(n) = 1 + 9*(9*e)^(1/3)*(Gamma(11/3) - Gamma(11/3, 1/3)). - Amiram Eldar, Dec 23 2022

A371077 Square array read by ascending antidiagonals: A(n, k) = 3^n*Pochhammer(k/3, n).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 28, 10, 3, 1, 0, 280, 80, 18, 4, 1, 0, 3640, 880, 162, 28, 5, 1, 0, 58240, 12320, 1944, 280, 40, 6, 1, 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1, 0, 24344320, 4188800, 524880, 58240, 6160, 648, 70, 8, 1

Offset: 0

Werner Schulte and Peter Luschny, Mar 10 2024

			The array starts:
  [0] 1,    1,     1,     1,     1,      1,      1,      1,      1, ...
  [1] 0,    1,     2,     3,     4,      5,      6,      7,      8, ...
  [2] 0,    4,    10,    18,    28,     40,     54,     70,     88, ...
  [3] 0,   28,    80,   162,   280,    440,    648,    910,   1232, ...
  [4] 0,  280,   880,  1944,  3640,   6160,   9720,  14560,  20944, ...
  [5] 0, 3640, 12320, 29160, 58240, 104720, 174960, 276640, 418880, ...
.
Seen as the triangle T(n, k) = A(n - k, k):
  [0] 1;
  [1] 0,       1;
  [2] 0,       1,      1;
  [3] 0,       4,      2,     1;
  [4] 0,      28,     10,     3,    1;
  [5] 0,     280,     80,    18,    4,   1;
  [6] 0,    3640,    880,   162,   28,   5,  1;
  [7] 0,   58240,  12320,  1944,  280,  40,  6, 1;
  [8] 0, 1106560, 209440, 29160, 3640, 440, 54, 7, 1;
.
Illustrating the LU decomposition of A:
    / 1                \   / 1 1 1 1 1 ... \   / 1   1   1    1    1 ... \
    | 0   1            |   |   1 2 3 4 ... |   | 0   1   2    3    4 ... |
    | 0   4   2        | * |     1 3 6 ... | = | 0   4  10   18   28 ... |
    | 0  28  24   6    |   |       1 4 ... |   | 0  28  80  162  280 ... |
    | 0 280 320 144 24 |   |         1 ... |   | 0 280 880 1944 3640 ... |
    | . . .            |   | . . .         |   | . . .                   |

Paolo Xausa, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals, flattened).

Family m^n*Pochhammer(k/m, n): A094587 (m=1), A370419 (m=2), this sequence (m=3), A370915 (m=4).
Rows: A000012, A001477, A028552.
Columns: A000007, A007559, A008544, A032031, A007559 (shifted), A034000, A034001, A051604, A051605, A051606, A051607, A051608, A051609.
Cf. A303486 (main diagonal), A371079 (row sums of triangle), A371076, A371080.

A := (n, k) -> 3^n*pochhammer(k/3, n):
A := (n, k) -> local j; mul(3*j + k, j = 0..n-1):
# Read by antidiagonals:
T := (n, k) -> A(n - k, k): seq(seq(T(n, k), k = 0..n), n = 0..9);
seq(lprint([n], seq(T(n, k), k = 0..n)), n = 0..9);
# Using the generating polynomials of the rows:
P := (n, x) -> local k; add(Stirling1(n, k)*(-3)^(n - k)*x^k, k=0..n):
seq(lprint([n], seq(P(n, k), k = 0..9)), n = 0..5);
# Using the exponential generating functions of the columns:
EGFcol := proc(k, len) local egf, ser, n; egf := (1 - 3*x)^(-k/3);
ser := series(egf, x, len+2): seq(n!*coeff(ser, x, n), n = 0..len) end:
seq(lprint([k], EGFcol(k, 8)), k = 0..6);
# As a matrix product:
with(LinearAlgebra):
L := Matrix(7, 7, (n, k) -> A371076(n - 1,  k - 1)):
U := Matrix(7, 7, (n, k) -> binomial(n - 1, k - 1)):
MatrixMatrixMultiply(L, Transpose(U));

Table[3^(n-k)*Pochhammer[k/3, n-k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Mar 14 2024 *)

def A(n, k): return 3**n * rising_factorial(k/3, n)
def A(n, k): return (-3)**n * falling_factorial(-k/3, n)

A(n, k) = Product_{j=0..n-1} (3*j + k).
A(n, k) = A(n+1, k-3) / (k - 3) for k > 3.
A(n, k) = Sum_{j=0..n} Stirling1(n, j)*(-3)^(n - j)* k^j.
A(n, k) = k! * [x^k] (exp(x) * p(n, x)), where p(n, x) are the row polynomials of A371080.
E.g.f. of column k: (1 - 3*t)^(-k/3).
E.g.f. of row n: exp(x) * (Sum_{k=0..n} A371076(n, k) * x^k / (k!)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n / (n!) = 1/(1 - x/(1 - 3*t)^(1/3)).
Sum_{n>=0, k>=0} A(n, k) * x^k * t^n /(n! * k!) = exp(x/(1 - 3*t)^(1/3)).
The LU decomposition of this array is given by the upper triangular matrix U which is the transpose of A007318 and the lower triangular matrix L = A371076, i.e., A(n, k) = Sum_{i=0..k} A371076(n, i) * binomial(k, i).

cp's OEIS Frontend

A034000 One half of triple factorial numbers.

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A257620 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 3*n + 3.

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A051604 a(n) = (3*n+4)!!!/4!!!.

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A051605 a(n) = (3*n+5)!!!/5!!!.

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A051606 a(n) = (3*n+6)!!!/6!!!, related to A032031 ((3*n)!!! triple factorials).

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A051609 a(n) = (3*n+9)!!!/9!!!, related to A032031 ((3*n)!!! triple factorials).

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A303486 a(n) = n! * [x^n] 1/(1 - 3*x)^(n/3).

A257620 Triangle read by rows: T(n, k) = t(n-k, k), where 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), and f(n) = 3*n + 3.

A051606 a(n) = (3n+6)!!!/6!!!, related to A032031 ((3n)!!! triple factorials).

A051609 a(n) = (3n+9)!!!/9!!!, related to A032031 ((3n)!!! triple factorials).