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 A136221 Column 0 of triangles A136220 and A136228; also equals column 0 of tables A136217 and A136218. 11
 1, 1, 3, 15, 108, 1036, 12569, 185704, 3247546, 65762269, 1515642725, 39211570981, 1125987938801, 35554753133312, 1224882431140838, 45731901253649898, 1839804317195739634, 79355626796692509253, 3653687500034925338348 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..90 EXAMPLE 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. PROG (PARI) /* 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]} (PARI) /* 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]} CROSSREFS 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. Sequence in context: A245835 A362354 A090351 * A153305 A110328 A217061 Adjacent sequences: A136218 A136219 A136220 * A136222 A136223 A136224 KEYWORD nonn AUTHOR Paul D. Hanna, Dec 25 2007 STATUS approved

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Last modified September 28 14:37 EDT 2023. Contains 365735 sequences. (Running on oeis4.)