A136220 Triangle P, read by rows, where column k of P^3 equals column 0 of P^(3k+3) such that column 0 of P^3 equals column 0 of P shift one row up, with P(0,0)=1.

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, 3247546, 2647628, 742620, 122350, 14330, 1325, 105, 8, 1, 65762269, 55251994, 15624420, 2571620

Offset: 0

Author

Paul D. Hanna, Dec 25 2007, corrected Jan 24 2008

Keywords

• nice
• nonn
• tabl

Examples

			Triangle P begins:
         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;
   3247546,  2647628,   742620,  122350,  14330,  1325,  105,   8, 1;
  65762269, 55251994, 15624420, 2571620, 298240, 26943, 2030, 136, 9, 1; ...
where column k of P = column 0 of U^(k+1) and U = A136228.
Matrix cube, W = P^3 (A136231), begins:
       1;
       3,     1;
      15,     6,     1;
     108,    48,     9,    1;
    1036,   495,    99,   12,   1;
   12569,  6338,  1323,  168,  15,  1;
  185704, 97681, 21036, 2754, 255, 18, 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 row up.
Matrix square, P^2 (A136225), begins:
      1;
      2,     1;
      8,     4,    1;
     49,    26,    6,    1;
    414,   232,   54,    8,   1;
   4529,  2657,  629,   92,  10,  1;
  61369, 37405, 9003, 1320, 140, 12, 1; ...
where column k of P^2 = column 0 of V^(k+1) and
triangle V = A136230 begins:
      1;
      2,     1;
      8,     5,     1;
     49,    35,     8,    1;
    414,   325,    80,   11,   1;
   4529,  3820,   988,  143,  14,  1;
  61369, 54800, 14696, 2200, 224, 17, 1; ...
where column k of V = column 0 of P^(3k+2).
Related triangle U = A136228 begins:
      1;
      1,     1;
      3,     4,    1;
     15,    24,    7,    1;
    108,   198,   63,   10,   1;
   1036,  2116,  714,  120,  13,  1;
  12569, 28052, 9884, 1725, 195, 16, 1; ...
where column k of U = column 0 of P^(3k+1)
and column k of P = column 0 of U^(k+1).
Surprisingly, column 0 of P is also found in square A136217:
(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),...;
(3),(8),15,(24),34,(46),59,(74),90,108,(127),147,169,(192),...;
(15),(49),108,(198),306,(453),622,(838),1080,1377,(1704),...;
(108),(414),1036,(2116),3493,(5555),8040,(11477),15483,...;
(1036),(4529),12569,(28052),48800,(82328),124335,(186261),...;
(12569),(61369),185704,(446560),811111,(1438447),2250731,...;
...
and has a recurrence similar to that of square array A136212
which generates the triple factorials.
		

Crossrefs

Columns: A136221, A136222, A136223, A136224.

Related tables: A136228 (U), A136230 (V), A136231 (W=P^3), A136217, A136218.

Variants: A091351, A135880.

Programs

PARI

{T(n,k)=local(P=Mat(1),U,PShR);if(n>0,for(i=0,n, PShR=matrix(#P,#P, r,c,
if(r>=c,if(r==c,1,if(c==1,0,P[r-1,c-1]))));U=P*PShR^2; U=matrix(#P+1,
#P+1, r,c, if(r>=c, if(r<#P+1,U[r,c], if(c==1,(P^3)[ #P,1],(P^(3*c-1))[r-c+1,
1])))); P=matrix(#U, #U, r,c, if(r>=c, if(r<#R,P[r,c], (U^c)[r-c+1,
1])))));P[n+1,k+1]}

Formula

Denote this triangle by P and define as follows.

Let [P^m]_k denote column k of matrix power P^m,

so that triangular matrix W = A136231 may be defined by

[W]_k = [P^(3k+3)]_0, for k>=0, such that

(1) W = P^3 and (2) [W]_0 = [P]_0 shift up one row.

Define the triangular matrix U = A136228 by

[U]_k = [P^(3k+1)]_0, for k>=0,

and the triangular matrix V = A136230 by

[V]_k = [P^(3k+2)]_0, for k>=0.

Then columns of P may be formed from powers of U:

[P]_k = [U^(k+1)]_0, for k>=0,

and columns of P^2 may be formed from powers of V:

[P^2]_k = [V^(k+1)]_0, for k>=0.

Further, columns of powers of P, U, V and W satisfy:

[U^(j+1)]_k = [P^(3k+1)]_j,

[V^(j+1)]_k = [P^(3k+2)]_j,

[W^(j+1)]_k = [P^(3k+3)]_j,

[W^(j+1)]_k = [W^(k+1)]_j,

[P^(3j+3)]_k = [P^(3k+3)]_j, for all j>=0, k>=0.

Also, we have the column transformations:

U * [P]k = [P]{k+1},

V * [P^2]k = [P^2]{k+1},

W * [P^3]k = [P^3]{k+1},

W * [U]k = [U]{k+1},

W * [V]k = [V]{k+1},

W * [W]k = [W]{k+1}, for all k>=0.

Other identities include the matrix products:

U = P * [P^2 shift right one column];

V = P^2 * [P shift right one column];

V = U * [U shift down one row];

W = V * [V shift down one row];

where the triangle transformations "shift right" and "shift down" are illustrated in examples of entries A136228 (U) and A136230 (V).

Extensions

Typo in example corrected by Paul D. Hanna, Mar 27 2010