A136217 -id:A136217 - cp's OEIS frontend

A136221 Column 0 of triangles A136220 and A136228; also equals column 0 of tables A136217 and A136218.

1, 1, 3, 15, 108, 1036, 12569, 185704, 3247546, 65762269, 1515642725, 39211570981, 1125987938801, 35554753133312, 1224882431140838, 45731901253649898, 1839804317195739634, 79355626796692509253, 3653687500034925338348

Offset: 0

Paul D. Hanna, Dec 25 2007

nonn

P = A136220 is a triangular matrix 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 place left. Tables A136217 and A136218 are defined by recurrences seemingly unrelated to the matrix product recurrence of A136220 and yet they all generate this same sequence in column 0.

			Equals column 0 of triangle P=A136220, which 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^3 = column 0 of P^(3k+3) such that
column 0 of P^3 = column 0 of P shift one place left.
Surprisingly, column 0 of P is also found in square A136218:
(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.

Paul D. Hanna, Table of n, a(n) for n = 0..90

Cf. A136220 (P), A136228 (U), A136231 (W=P^3).
Cf. other columns of P: A136222, A136223, A136224.
Cf. related tables: A136217, A136218.
Cf. variants: A091352, A135881.

/* Generate using matrix product recurrences of triangle 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[n+1,1]}

/* Generated as column 0 in triangle A136218 (faster): */ {a(n)=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)))))));A[1]}

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.

A136229 Column 1 of triangle A136228; also equals column 3 of square array A136217.

Original entry on oeis.org

1, 4, 24, 198, 2116, 28052, 446560, 8325700, 178284892, 4317391240, 116757568072, 3489919886180, 114304360656596, 4072431454706374, 156845310426959972, 6494800764749659199, 287796117221639178946, 13590112496057829388516

Offset: 0

Paul D. Hanna, Jan 28 2008

nonn

Cf. A136228, A136217; A136221.

{a(n)=local(P=Mat(1),U=Mat([1,0;1,1]),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])))));U[n+2,2]}

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.

A136230 Triangle V, read by rows, where column k of V^(j+1) = column j of P^(3k+2), for j>=0, k>=0 and where P=A136220.

Original entry on oeis.org

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, 996815, 932761, 257264, 39468, 4123, 323, 20, 1, 18931547, 18426632, 5198680, 812801, 86506, 6919, 440, 23, 1, 412345688

Offset: 0

Paul D. Hanna, Jan 28 2008

			This triangle V 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;
996815, 932761, 257264, 39468, 4123, 323, 20, 1;
18931547, 18426632, 5198680, 812801, 86506, 6919, 440, 23, 1; ...
where column k of V = column 0 of P^(3k+2) 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^2 = column 0 of V^(k+1).
Also, this triangle V equals the matrix product:
V = P^2 * [P shift right one column]
where 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; ...
and P shift right one column begins:
1;
0, 1;
0, 1, 1;
0, 3, 2, 1;
0, 15, 10, 3, 1;
0, 108, 75, 21, 4, 1;
0, 1036, 753, 208, 36, 5, 1; ...
Also, this triangle V equals the matrix product:
V = U * [U shift down one row]
where 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; ...
and U shift down one row begins:
1;
1, 1;
1, 1, 1;
3, 4, 1, 1;
15, 24, 7, 1, 1;
108, 198, 63, 10, 1, 1;
1036, 2116, 714, 120, 13, 1, 1; ...

Cf. A136226 (column 0), A136229 (column 1); related tables: A136220 (P), A136225 (P^2), A136230 (V), A136231 (W=P^3), A136234 (V^2), A136237 (V^3); A136217, A136218.

{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[n+1,k+1]}

Triangle W=P^3=A136231 transforms column k of V into column k+1 of V. This triangle equals the matrix products: V = P^2 * [P shift right one column] and V = U * [U shift down one row] (see examples).

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]}

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.

Original entry on oeis.org

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("")) */

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).

cp's OEIS Frontend

A136221 Column 0 of triangles A136220 and A136228; also equals column 0 of tables A136217 and A136218.

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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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A136229 Column 1 of triangle A136228; also equals column 3 of square array A136217.

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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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A136230 Triangle V, read by rows, where column k of V^(j+1) = column j of P^(3k+2), for j>=0, k>=0 and where 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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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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