A136230 -id:A136230 - cp's OEIS frontend

A136227 Column 1 of triangle A136225; also equals column 0 of triangle A136230.

1, 4, 26, 232, 2657, 37405, 627435, 12248365, 273211787, 6862775083, 191840407156, 5909873159107, 199002812894375, 7273866200397039, 286882936292798852, 12145886485652450131, 549504341899436759416

Offset: 0

Paul D. Hanna, Jan 28 2008

nonn

Equals column 1 of P^2 (A136225) and equals column 0 of V^2, where P = A136220 and V = A136230 are triangular matrices such that column k of V = column 0 of P^(3k+2) and column j of P^2 = column 0 of V^(j+1).

Cf. A136226, A136225 (P^2), A136220 (P), A136230 (V); A136217.

/* Generate using matrix product recurrences of triangle P=A136220: */ {a(n)=local(P=Mat([1,0;1,1]),U,PShR);if(n>0,for(i=0,n+1, 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^2)[n+2,2]}

A136234 Matrix square of triangle V = A136230, read by rows.

Original entry on oeis.org

1, 4, 1, 26, 10, 1, 232, 110, 16, 1, 2657, 1435, 248, 22, 1, 37405, 22135, 4240, 440, 28, 1, 627435, 397820, 81708, 9295, 686, 34, 1, 12248365, 8203057, 1773156, 214478, 17248, 986, 40, 1, 273211787, 191405232, 43039532, 5442349, 463267, 28747, 1340

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, V^2, begins:
1;
4, 1;
26, 10, 1;
232, 110, 16, 1;
2657, 1435, 248, 22, 1;
37405, 22135, 4240, 440, 28, 1;
627435, 397820, 81708, 9295, 686, 34, 1;
12248365, 8203057, 1773156, 214478, 17248, 986, 40, 1;
273211787, 191405232, 43039532, 5442349, 463267, 28747, 1340, 46, 1; ...
where column 0 of V^2 = column 1 of P^2 = triangle A136225.

Cf. A136227 (column 0); related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W=P^3), A136237 (V^3).

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

Column k of V^2 (this triangle) = column 1 of P^(3k+2), where P = triangle A136220.

A136237 Matrix cube of triangle V = A136230, read by rows.

Original entry on oeis.org

1, 6, 1, 54, 15, 1, 629, 225, 24, 1, 9003, 3770, 504, 33, 1, 153276, 71655, 10988, 891, 42, 1, 3031553, 1539315, 259236, 23903, 1386, 51, 1, 68406990, 37072448, 6688092, 672672, 44135, 1989, 60, 1, 1736020806, 992226060, 188767184, 20225436, 1442049

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, V^3, begins:
1;
6, 1;
54, 15, 1;
629, 225, 24, 1;
9003, 3770, 504, 33, 1;
153276, 71655, 10988, 891, 42, 1;
3031553, 1539315, 259236, 23903, 1386, 51, 1;
68406990, 37072448, 6688092, 672672, 44135, 1989, 60, 1;
1736020806, 992226060, 188767184, 20225436, 1442049, 73304, 2700, 69, 1;
where column 0 of V^3 = column 2 of P^2 = triangle A136225.

Cf. related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W=P^3), A136234 (V^2).

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

Column k of V^3 (this triangle) = column 2 of P^(3k+2), where P = triangle A136220.

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.

Original entry on oeis.org

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

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

			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.

Columns: A136221, A136222, A136223, A136224.
Related tables: A136228 (U), A136230 (V), A136231 (W=P^3), A136217, A136218.
Variants: A091351, A135880.

{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]}

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).

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

A136228 Triangle U, read by rows, where column k of U^(j+1) = column j of P^(3k+1) for j>=0, k>=0 and P=A136220.

Original entry on oeis.org

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, 185704, 446560, 162729, 29190, 3393, 288, 19, 1, 3247546, 8325700, 3117660, 571225, 67756, 5880, 399, 22, 1, 65762269, 178284892

Offset: 0

Paul D. Hanna, Jan 28 2008

			Triangle U 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;
185704, 446560, 162729, 29190, 3393, 288, 19, 1;
3247546, 8325700, 3117660, 571225, 67756, 5880, 399, 22, 1; ...
where column k of U = column 0 of P^(3k+1) and
triangle P = A136220 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; ...
where column k of P = column 0 of U^(k+1).
Also, this triangle U can be obtained by the matrix product:
U = P * [P^2 shift right one column]
where P^2 shift right one column begins:
1;
0, 1;
0, 2, 1;
0, 8, 4, 1;
0, 49, 26, 6, 1;
0, 414, 232, 54, 8, 1;
0, 4529, 2657, 629, 92, 10, 1;
0, 61369, 37405, 9003, 1320, 140, 12, 1; ...

Cf. A136221 (column 0), A136229 (column 1); related tables: A136220 (P), A136225 (P^2), A136230 (V), A136231 (W=P^3), A136217, A136218.

{T(n,k)=local(P=Mat(1),U=Mat(1),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])))));U[n+1,k+1]}

This triangle U = P * [P^2 shift right one column] (see example), where P = A136220 and P^2 = A136225.

A136231 Triangle W, read by rows, where column k of W = column 0 of W^(k+1) for k>=0 such that W equals the matrix cube of P = A136220 with column 0 of W = column 0 of P shift up one row.

Original entry on oeis.org

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, 3247546, 1767845, 390012, 52204, 4950, 360, 21, 1, 65762269, 36839663, 8287041, 1128404, 108860, 8073, 483, 24, 1, 1515642725

Offset: 0

Paul D. Hanna, Jan 28 2008

This triangle W is the column transform for triangles U=A136228 and V=A136230: W * [column k of U] = column k+1 of U and W * [column k of V] = column k+1 of V, for k>=0.

			Triangle W 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;
3247546, 1767845, 390012, 52204, 4950, 360, 21, 1;
65762269, 36839663, 8287041, 1128404, 108860, 8073, 483, 24, 1; ...
where column k of W = column 0 of W^(k+1) such that W = P^3
and triangle P = A136220 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; ...
where column k of P^3 = column 0 of P^(3k+3) such that
column 0 of P^3 = column 0 of P shift up one row.
Also, this triangle W equals the matrix product:
W = V * [V shift down one row]
where 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; ...
and V shift down one row begins:
1;
1, 1;
2, 1, 1;
8, 5, 1, 1;
49, 35, 8, 1, 1;
414, 325, 80, 11, 1, 1;
4529, 3820, 988, 143, 14, 1, 1; ...

Cf. A136221 (column 0); related tables: A136220 (P), A136225 (P^2), A136228 (U), A136230 (V), A136235 (W^2), A136238 (W^3); A136217, A136218.

{T(n,k)=local(P=Mat(1),U=Mat(1),W=Mat(1),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]))); W=P^3;));W[n+1,k+1]}

A136225 Matrix square of triangle A136220, read by rows.

Original entry on oeis.org

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, 996815, 627435, 153276, 22606, 2385, 198, 14, 1, 18931547, 12248365, 3031553, 450066, 47500, 3904, 266, 16, 1, 412345688, 273211787

Offset: 0

Paul D. Hanna, Jan 28 2008

Column 0 of this triangle = column 1 of square array A136217.

			Let P = A136220, then this triangle is P^2 and 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;
996815, 627435, 153276, 22606, 2385, 198, 14, 1;
18931547, 12248365, 3031553, 450066, 47500, 3904, 266, 16, 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).
Triangle P = A136220 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; ...
where column k of P = column 0 of U^(k+1) and U = A136228.

Cf. columns: A136226, A136227; related tables: A136228 (U), A136230 (V), A136231 (W=P^3), A136217, A136218.

{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^2)[n+1,k+1]}

Let P=A136220, V=A136230, then column k of P^2 (this triangle) = column 0 of V^(k+1) while column j of V = column 0 of P^(3j+2).

A136226 Column 0 of P^2 where triangle P = A136220; also equals column 1 of square array A136217.

Original entry on oeis.org

1, 2, 8, 49, 414, 4529, 61369, 996815, 18931547, 412345688, 10143253814, 278322514093, 8432315243347, 279689506725247, 10083429764179733, 392703359698462567, 16433405366965493214, 735484032071079495354

Offset: 0

Paul D. Hanna, Jan 28 2008

nonn

Cf. A136225 (P^2), A136220 (P), A136230 (V); A136217.

/* Generate using matrix product recurrences of triangle P=A136220: */ {a(n)=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^2)[n+1,1]}

Equals column 0 of triangle V = A136230, where column k of V = column 0 of P^(3k+2) such that column k of P^2 = column 0 of V^(k+1), for k>=0 and where P = A136220.

A136233 Matrix square of triangle U = A136228, read by rows.

Original entry on oeis.org

1, 2, 1, 10, 8, 1, 75, 76, 14, 1, 753, 888, 196, 20, 1, 9534, 12542, 3087, 370, 26, 1, 146353, 209506, 55552, 7320, 598, 32, 1, 2647628, 4058806, 1136975, 159645, 14235, 880, 38, 1, 55251994, 89706276, 26224597, 3856065, 364403, 24480, 1216, 44, 1

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, U^2, begins:
1;
2, 1;
10, 8, 1;
75, 76, 14, 1;
753, 888, 196, 20, 1;
9534, 12542, 3087, 370, 26, 1;
146353, 209506, 55552, 7320, 598, 32, 1;
2647628, 4058806, 1136975, 159645, 14235, 880, 38, 1;
55251994, 89706276, 26224597, 3856065, 364403, 24480, 1216, 44, 1; ...
where column 0 of U^2 = column 1 of P = A136220.

Cf. A136222 (column 0); related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W=P^3), A136236 (U^3).

{T(n,k)=local(P=Mat(1),U=Mat(1),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])))));(U^2)[n+1,k+1]}

Column k of U^2 (this triangle) = column 1 of P^(3k+1), where P = triangle A136220.

A136235 Matrix square of triangle W = A136231; also equals P^6, where P = triangle A136220.

Original entry on oeis.org

1, 6, 1, 48, 12, 1, 495, 150, 18, 1, 6338, 2160, 306, 24, 1, 97681, 36103, 5643, 516, 30, 1, 1767845, 694079, 115917, 11592, 780, 36, 1, 36839663, 15164785, 2657946, 282122, 20655, 1098, 42, 1, 870101407, 372225541, 67708113, 7502470, 580780

Offset: 0

Paul D. Hanna, Feb 07 2008

			This triangle, W^2, begins:
1;
6, 1;
48, 12, 1;
495, 150, 18, 1;
6338, 2160, 306, 24, 1;
97681, 36103, 5643, 516, 30, 1;
1767845, 694079, 115917, 11592, 780, 36, 1;
36839663, 15164785, 2657946, 282122, 20655, 1098, 42, 1;
870101407, 372225541, 67708113, 7502470, 580780, 33480, 1470, 48, 1; ...
where column 0 of W^2 = column 1 of W = triangle A136231.

Cf. A136221 (column 0); related tables: A136220 (P), A136228 (U), A136230 (V), A136231 (W), A136238 (W^3).

{T(n,k)=local(P=Mat(1),U=Mat(1),W=Mat(1),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]))); W=P^3;));(W^2)[n+1,k+1]}

Column k of W^2 (this triangle) = column 1 of W^(k+1), where W = P^3 and P = triangle A136220.

cp's OEIS Frontend

A136227 Column 1 of triangle A136225; also equals column 0 of triangle A136230.

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A136234 Matrix square of triangle V = A136230, read by rows.

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A136237 Matrix cube of triangle V = A136230, read by rows.

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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.

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A136228 Triangle U, read by rows, where column k of U^(j+1) = column j of P^(3k+1) for j>=0, k>=0 and P=A136220.

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A136231 Triangle W, read by rows, where column k of W = column 0 of W^(k+1) for k>=0 such that W equals the matrix cube of P = A136220 with column 0 of W = column 0 of P shift up one row.

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A136225 Matrix square of triangle A136220, read by rows.

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A136226 Column 0 of P^2 where triangle P = A136220; also equals column 1 of square array A136217.

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A136233 Matrix square of triangle U = A136228, read by rows.

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