login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 19
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..48.

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).

EXAMPLE

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.

PROG

(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]}

CROSSREFS

Columns: A136221, A136222, A136223, A136224.

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

Variants: A091351, A135880.

Sequence in context: A135876 A136217 A166884 * A248035 A088956 A106208

Adjacent sequences:  A136217 A136218 A136219 * A136221 A136222 A136223

KEYWORD

nice,nonn,tabl

AUTHOR

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

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 1 05:29 EDT 2020. Contains 333155 sequences. (Running on oeis4.)