A136213 -id:A136213 - cp's OEIS frontend

A136212 Triple factorial array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {[m*(m+5)/6], m >= 0} and then taking partial sums, starting with all 1's in row 0.

1, 1, 1, 4, 2, 1, 28, 10, 3, 1, 280, 80, 18, 4, 1, 3640, 880, 162, 28, 5, 1, 58240, 12320, 1944, 280, 39, 6, 1, 1106560, 209440, 29160, 3640, 418, 52, 7, 1, 24344320, 4188800, 524880, 58240, 5714, 600, 66, 8, 1, 608608000, 96342400, 11022480, 1106560, 95064

Offset: 0

Paul D. Hanna, Dec 22 2007

This is the triple factorial variant of Moessner's factorial array described by A125714 and also of the double factorial array A135876. Another very interesting variant is A136217.

			Square array begins:
(1),(1),(1),1,(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,1,(1),1,1,1,...;
(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),19,20,21,..;
(4),(10),(18),28,(39),52,(66),82,(99),118,138,(159),182,206,(231),258,286,..;
(28),(80),(162),280,(418),600,(806),1064,(1350),1696,2074,(2485),2966,3484,..;
(280),(880),(1944),3640,(5714),8680,(12164),16840,(22194),29080,36824,(45474),.;
(3640),(12320),(29160),58240,(95064),151200,(219108),315440,(428652),581680,...;
(58240),(209440),(524880),1106560,(1864456),3082240,...;
where terms in parenthesis are at positions {[m*(m+5)/6], m>=0}
and are removed before taking partial sums to obtain the next row.
To generate the array, start with all 1's in row 0; from then on,
obtain row n+1 from row n by first removing terms in row n at
positions {[m*(m+5)/6], m>=0} and then taking partial sums.
For example, to generate row 2 from row 1:
[(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),...],
remove terms at positions [0,1,2,4,6,8,11,14,17,...] to get:
[4, 6, 8, 10,11, 13,14, 16,17, 19,20,21, 23,24,25, 27,28,29, ...]
then take partial sums to obtain row 2:
[4, 10, 18, 28,39, 52,66, 82,99, 118,138,159, 182,206,231, ...].
Continuing in this way will generate all the rows of this array.

Cf. columns: A007559, A008544, A032031, A024216, A024395; related tables: A136213, A125714, A135876, A127054, A125781; A136217.

t[n_, k_] := t[n, k] = Module[{a = 0, m = 0, c = 0, d = 0}, If[n == 0, a = 1, While[d <= k, If[c == Quotient[(m*(m + 5)), 6], m += 1, a += t[n - 1, c]; d += 1]; c += 1]]; a]; Table[t[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013, translated from Pari *)

{T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==(m*(m+5))\6, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}

Columns 0, 1 and 2 form the triple factorials A007559, A008544 and A032031, respectively. Column 4 equals A024216; column 6 equals A024395.

A136218 Triangle, read by rows of A136219(n) terms, where row n+1 is generated from row n by first inserting zeros in row n at positions {[m*(m+7)/6], m>=0} and then taking partial sums, starting with a '1' in row 0.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 2, 2, 1, 1, 15, 15, 15, 12, 12, 9, 9, 6, 6, 4, 2, 2, 1, 108, 108, 108, 93, 93, 78, 78, 63, 63, 51, 39, 39, 30, 21, 21, 15, 9, 9, 5, 3, 1, 1, 1036, 1036, 1036, 928, 928, 820, 820, 712, 712, 619, 526, 526, 448, 370, 370, 307, 244, 244, 193, 154, 115, 115, 85

Offset: 0

Paul D. Hanna, Dec 23 2007

A variant of the triple factorial triangle A136213. Compare to square array A136217, which is generated by a complementary process.

			Triangle begins:
1;
1,1,1;
3,3,3,2,2,1,1;
15,15,15,12,12,9,9,6,6,4,2,2,1;
108,108,108,93,93,78,78,63,63,51,39,39,30,21,21,15,9,9,5,3,1,1;
1036,1036,1036,928,928,820,820,712,712,619,526,526,448,370,370,307,244,244,193,154,115,115,85,64,43,43,28,19,10,10,5,2,1;
12569,12569,12569,11533,11533,10497,10497,9461,9461,8533,7605,7605,6785,5965,5965,5253,4541,4541,3922,3396,2870,2870,2422,2052,1682,1682,1375,1131,887,887,694,540,425,310,310,225,161,118,75,75,47,28,18,8,8,3,1;
...
Number of terms in rows is given by A136219, which starts:
[1,3,7,13,22,33,47,64,84,106,131,159,190,224,261,301,343,388,...].
To generate row 3, start with row 2:
[3,3,3,2,2,1,1];
insert zeros at positions [0,1,3,5,7,10] to get:
[0,0,3,0,3,0,3,0,2,2,0,1,1],
then take reverse partial sums (from right to left) to obtain row 3:
[15,15,15,12,12,9,9,6,6,4,2,2,1].
For row 4, insert zeros in row 3 at positions [0,1,3,5,7,10,13,16,20]:
[0,0,15,0,15,0,15,0,12,12,0,9,9,0,6,6,0,4,2,2,0,1]
then take reverse partial sums to obtain row 4:
[108,108,108,93,93,78,78,63,63,51,39,39,30,21,21,15,9,9,5,3,1,1].
Continuing in this way will generate all the rows of this triangle.
Amazingly, column 0 of this triangle = column 0 of triangle P=A136220:
1;
1, 1;
3, 2, 1;
15, 10, 3, 1;
108, 75, 21, 4, 1;
1036, 753, 208, 36, 5, 1;
12569, 9534, 2637, 442, 55, 6, 1;
185704, 146353, 40731, 6742, 805, 78, 7, 1; ...
where column k of P^3 = column 0 of P^(3k+3) such that
column 0 of P^3 = column 0 of P shift one place left.

Cf. A136221 (column 0), A136219; A136213, A136220.

{T(n,k)=local(A=[1],B);if(n>0,for(i=1,n,m=1;B=[0]; for(j=1,#A,if(j+m-1==(m*(m+7))\6,m+=1;B=concat(B,0));B=concat(B,A[j])); A=Vec(Polrev(Vec(Pol(B)/(1-x+O(x^#B)))))));if(k+1>#A,0,A[k+1])} /* for(n=0,6,for(k=0,2*n^2,if(T(n,k)==0,break,print1(T(n,k),",")));print("")) */

A136214 Triangle U, read by rows, where U(n,k) = Product_{j=k..n-1} (3*j+1) for n > k with U(n,n) = 1.

Original entry on oeis.org

1, 1, 1, 4, 4, 1, 28, 28, 7, 1, 280, 280, 70, 10, 1, 3640, 3640, 910, 130, 13, 1, 58240, 58240, 14560, 2080, 208, 16, 1, 1106560, 1106560, 276640, 39520, 3952, 304, 19, 1, 24344320, 24344320, 6086080, 869440, 86944, 6688, 418, 22, 1

Offset: 0

Paul D. Hanna, Feb 07 2008

Let G(m, k, p) = (-p)^k*Product_{j=0..k-1}(j - m - 1/p) and T(n, k, p) = G(n-1, n-k, p) then T(n, k, 1) = A094587(n, k), T(n, k, 2) = A112292(n, k) and T(n, k, 3) is this sequence. - Peter Luschny, Jun 01 2009, revised Jun 18 2019

			Triangle begins:
        1;
        1,       1;
        4,       4,      1;
       28,      28,      7,     1;
      280,     280,     70,    10,    1;
     3640,    3640,    910,   130,   13,   1;
    58240,   58240,  14560,  2080,  208,  16,  1;
  1106560, 1106560, 276640, 39520, 3952, 304, 19, 1; ...
Matrix inverse begins:
   1;
  -1,   1;
   0,  -4,   1;
   0,   0,  -7,   1;
   0,   0,   0, -10,   1;
   0,   0,   0,   0, -13,   1; ...

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Cf. A094587, A112333, A136216, A136239; A007559, A136212, A136213.

[[n eq 0 select 1 else k eq n select 1 else (&*[3*j+1: j in [k..n-1]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 14 2019

nmax:=8; for n from 0 to nmax do U(n, n):=1 od: for n from 0 to nmax do for k from 0 to n do if n > k then U(n, k) := mul((3*j+1), j = k..n-1) fi: od: od: for n from 0 to nmax do seq(U(n, k), k=0..n) od: seq(seq(U(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jul 04 2011, revised Nov 23 2012

Table[Product[3*j+1, {j,k,n-1}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 14 2019 *)

PARI
```
T(n,k)=if(n==k,1,prod(j=k,n-1,3*j+1))
```

def T(n, k):
    if (k==n): return 1
    else: return product(3*j+1 for j in (k..n-1))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 14 2019

Matrix powers: column 0 of U^(k+1) = column k of A136216 for k >= 0; simultaneously, column k = column 0 of A136216^(3k+1) for k >= 0. Element in column 0, row n, of matrix power U^(k+1) = A007559(n)*C(n+k,k), where A007559 are triple factorials found in column 0 of this triangle.

A136215 Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.

Original entry on oeis.org

1, 1, 1, 4, 2, 1, 28, 12, 3, 1, 280, 112, 24, 4, 1, 3640, 1400, 280, 40, 5, 1, 58240, 21840, 4200, 560, 60, 6, 1, 1106560, 407680, 76440, 9800, 980, 84, 7, 1, 24344320, 8852480, 1630720, 203840, 19600, 1568, 112, 8, 1, 608608000, 219098880, 39836160

Offset: 0

Paul D. Hanna, Feb 07 2008

Comments from Peter Bala, Jul 10 2008: (Start) This array is the particular case P(1,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown below
n\k|0....................1...............2.........3.....4
----------------------------------------------------------
0..|1.....................................................
1..|a....................1................................
2..|a(a+b)...............2a..............1................
3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
...
See A094587 for some general properties of these arrays.
Other cases recorded in the database include: P(1,0) = Pascal's triangle A007318, P(1,1) = A094587, P(2,0) = A038207, P(3,0) = A027465, P(2,1) = A132159 and P(2,3) = A136216. (End)
The generalized Pascal matrix that Bala refers to is itself a special case of application of the formalism of A133314 to fundamental matrices derived from infinitesimal generators described in A133314, of which the fundamental Pascal (A007318), unsigned Lah (A105278) and associated Laguerre (A135278) matrices are special examples. The formalism gives, among other relations, the inverse of T as TI(n,k) = b(n-k)*C(n,k) where the sequence b is given by the list partition transform (A133314) of A007559; i.e., b = LPT(A007559) = (1,-A008544)= (1,-1,-2,-10,-80,...). The formalism of A132382 may also be applied with the double factorial A001147 replaced by the triple factorial A007559 (see also A133480). - Tom Copeland, Aug 18 2008
From Peter Bala, Aug 29 2013: (Start)
Exponential Riordan array [1/(1 - 3*y)^(1/3), y]. The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x).
For example, row 3 is (28, 12, 3, 1) so P(3,x + 1) = (x + 1)*(x + 4)*(x + 7) = 28 + 12*x + 3*x*(x + 3) + x*(x + 3)*(x + 6). (End)

			Column k of T = column 0 of U^(k+1), while
column k of U = column 0 of T^(3k+1) where U = A136214 and
column k of V = column 0 of T^(3k+2) where V = A112333.
This triangle T begins:
        1;
        1,      1;
        4,      2,     1;
       28,     12,     3,    1;
      280,    112,    24,    4,   1;
     3640,   1400,   280,   40,   5,  1;
    58240,  21840,  4200,  560,  60,  6, 1;
  1106560, 407680, 76440, 9800, 980, 84, 7, 1; ...
Triangle U = A136214 begins:
     1;
     1,    1;
     4,    4,   1;
    28,   28,   7,   1;
   280,  280,  70,  10,  1;
  3640, 3640, 910, 130, 13, 1; ...
with triple factorials A007559 in column 0.
Triangle V = A112333 begins:
      1;
      2,    1;
     10,    5,    1;
     80,   40,    8,   1;
    880,  440,   88,  11,  1;
  12320, 6160, 1232, 154, 14, 1; ...
with triple factorials A008544 in column 0.

G. C. Greubel, Rows n=0..100 of triangle, flattened
M. Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3
Wikipedia, Sheffer sequence

Cf. A136216 (matrix square); A007559, A008544; A136212, A136213.

Cf. A094587.

T[n_, k_]:= Binomial[n, k]*If[n - k == 0, 1, Product[3*j + 1, {j, 0, n - k - 1}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 10 2018 *)

T(n,k)=binomial(n,k)*if(n-k==0,1,prod(j=0,n-k-1,3*j+1))

Column k of T = column 0 of U^(k+1) (matrix power) for k>=0 where U = A136214. Matrix square equals A136216, where A136216(n,k) = A008544(n-k)*C(n,k) where A008544 are also triple factorials.
From Peter Bala, Jul 10 2008: (Start)
T(n,k) = (3*n-3*k-2)*T(n-1,k) + T(n-1,k-1).
E.g.f. exp(x*y)/(1-3*y)^(1/3) = 1 + (1+x)*y + (4+2*x+x^2)*y^2/2! + ... . (End)

A136216 Triangle T, read by rows, where T(n,k) = A008544(n-k)*C(n,k) where A008544 equals the triple factorials in column 0.

Original entry on oeis.org

1, 2, 1, 10, 4, 1, 80, 30, 6, 1, 880, 320, 60, 8, 1, 12320, 4400, 800, 100, 10, 1, 209440, 73920, 13200, 1600, 150, 12, 1, 4188800, 1466080, 258720, 30800, 2800, 210, 14, 1, 96342400, 33510400, 5864320, 689920, 61600, 4480, 280, 16, 1

Offset: 0

Paul D. Hanna, Feb 07 2008

This array is the particular case P(2,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown in the comments to A094587. - Peter Bala, Jul 10 2008

The row polynomials form an Appell sequence. - Tom Copeland, Dec 03 2013

			Triangle begins:
1;
2, 1;
10, 4, 1;
80, 30, 6, 1;
880, 320, 60, 8, 1;
12320, 4400, 800, 100, 10, 1;
209440, 73920, 13200, 1600, 150, 12, 1;
4188800, 1466080, 258720, 30800, 2800, 210, 14, 1; ...

Wikipedia, Appell sequence
Wikipedia, Sheffer sequence

Cf. A136215 (square-root), A112333, A008544, A136212, A136213.

Cf. A094587.

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1/(1 - 3 #)^(2/3)&, #&, 9, True] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

{T(n,k) = binomial(n,k)*if(n-k==0,1, prod(j=0,n-k-1,3*j+2))}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Column k of T = column 0 of V^(k+1) for k>=0 where V = A112333.
Equals the matrix square of triangle A136215.
T(n,k) = (3*n-3*k-1)*T(n-1,k) + T(n-1,k-1). - Peter Bala, Jul 10 2008
Using the formalism of A132382 modified for the triple rather than the double factorial (replace 2 by 3 in basic formulas), the e.g.f. for the row polynomials is exp(x*t)*(1-3x)^(-2/3). - Tom Copeland, Aug 18 2008
From Peter Bala, Aug 28 2013: (Start)
Exponential Riordan array [1/(1 - 3*y)^(2/3), y].
The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (2*x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x). For example, row 3 is (80, 30, 6, 1) so P(3,x + 1) = (2*x + 2)*(2*x + 5)*(2*x + 8) = 80 + 20*(2*x) + 6*(2*x*(2*x + 3)) + (2*x)*(2*x + 3)*(2*x + 6). (End)

A156628 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2, m>0} and then taking partial sums, starting with all 1's in row 0.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 13, 7, 3, 1, 71, 33, 13, 4, 1, 461, 191, 71, 20, 5, 1, 3447, 1297, 461, 120, 28, 6, 1, 29093, 10063, 3447, 836, 181, 38, 7, 1, 273343, 87669, 29093, 6616, 1333, 270, 49, 8, 1, 2829325, 847015, 273343, 58576, 11029, 2150, 375, 61, 9, 1

Offset: 0

Paul D. Hanna, Feb 17 2009

			To generate the array, start with all 1's in row 0; from then on,
obtain row n+1 from row n by first removing terms in row n at
positions 0 and {(m+1)*(m+2)/2-2,m>0} and then taking partial sums.
This square array A begins:
(1), (1), 1, 1, (1), 1, 1, 1, (1), 1, 1, 1, 1, (1), 1, 1, 1, 1, 1, ...;
(1), (2), 3, 4, (5), 6, 7, 8, (9), 10, 11, 12, 13, (14), 15, 16, ...;
(3), (7), 13, 20, (28), 38, 49, 61, (74), 89, 105, 122, 140, (159),...;
(13), (33), 71, 120, (181), 270, 375, 497, (637), 817, 1019, 1244, ...;
(71), (191), 461, 836, (1333), 2150, 3169, 4413, (5906), 8001, ...;
(461), (1297), 3447, 6616, (11029), 19030, 29483, 42775, (59324),...;
(3447), (10063), 29093, 58576, (101351), 185674, 300329, 451277, ...;
(29093), (87669), 273343, 573672, (1024949), 1982310, 3330651, ...;
(273343), (847015), 2829325, 6159976, (11320359), 23009602, 39998897, ...;
where terms in parenthesis at positions {0,1,4,8,13,..} in a row
are removed before taking partial sums to obtain the next row.
...
RELATION TO SPECIAL TRIANGLE.
Triangle A104980 begins:
1;
1, 1;
3, 2, 1;
13, 7, 3, 1;
71, 33, 13, 4, 1;
461, 191, 71, 21, 5, 1;
3447, 1297, 461, 133, 31, 6, 1;
29093, 10063, 3447, 977, 225, 43, 7, 1; ...
in which column 0 and column 1 are found in square array A.
...
Matrix square of A104980 = triangle A104988 which begins:
1;
2, 1;
8, 4, 1;
42, 20, 6, 1;
266, 120, 38, 8, 1;
1954, 836, 270, 62, 10, 1;
16270, 6616, 2150, 516, 92, 12, 1;
151218, 58576, 19030, 4688, 882, 128, 14, 1; ...
where column 1 and column 2 are also found in square array A.

Cf. columns: A003319, A104981, A156629, related triangles: A104980, A104988.

Cf. related tables: A136212, A136213, A125714, A135876, A127054, A125781, A136217.

{T (n, k)=local (A=0, b=2, c=1, d=0); if (n==0, A=1, until (d>k, if (c==b* (b+1)/2-2, b+=1, A+=T (n-1, c); d+=1); c+=1)); A}

Column 0 = Column 0 of triangle A104980 = A003319.
Column 1 = Column 1 of triangle A104980 = A104981.
Column 3 = column 1 of A104988 (matrix square of A104980).
Column 5 = column 2 of A104988 (matrix square of A104980).

cp's OEIS Frontend

A136212 Triple factorial array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {[m*(m+5)/6], m >= 0} and then taking partial sums, starting with all 1's in row 0.

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A136218 Triangle, read by rows of A136219(n) terms, where row n+1 is generated from row n by first inserting zeros in row n at positions {[m*(m+7)/6], m>=0} and then taking partial sums, starting with a '1' in row 0.

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A136214 Triangle U, read by rows, where U(n,k) = Product_{j=k..n-1} (3*j+1) for n > k with U(n,n) = 1.

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A136215 Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.

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A136216 Triangle T, read by rows, where T(n,k) = A008544(n-k)*C(n,k) where A008544 equals the triple factorials in column 0.

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A156628 Square array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions 0 and {(m+1)*(m+2)/2-2, m>0} and then taking partial sums, starting with all 1's in row 0.

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