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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 25 2007
STATUS
approved