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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. 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 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

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