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

%I A136221 #4 Mar 30 2012 18:37:08
%S A136221 1,1,3,15,108,1036,12569,185704,3247546,65762269,1515642725,
%T A136221 39211570981,1125987938801,35554753133312,1224882431140838,
%U A136221 45731901253649898,1839804317195739634,79355626796692509253,3653687500034925338348
%N A136221 Column 0 of triangles A136220 and A136228; also equals column 0 of tables A136217 and A136218.
%C A136221 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.
%H A136221 Paul D. Hanna, <a href="/A136221/b136221.txt">Table of n, a(n) for n = 0..90</a>
%e A136221 Equals column 0 of triangle P=A136220, which begins:
%e A136221 1;
%e A136221 1, 1;
%e A136221 3, 2, 1;
%e A136221 15, 10, 3, 1;
%e A136221 108, 75, 21, 4, 1;
%e A136221 1036, 753, 208, 36, 5, 1;
%e A136221 12569, 9534, 2637, 442, 55, 6, 1;
%e A136221 185704, 146353, 40731, 6742, 805, 78, 7, 1; ...
%e A136221 where column k of P^3 = column 0 of P^(3k+3) such that
%e A136221 column 0 of P^3 = column 0 of P shift one place left.
%e A136221 Surprisingly, column 0 of P is also found in square A136218:
%e A136221 (1),(1),1,(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,1,(1),...;
%e A136221 (1),(2),3,(4),5,(6),7,(8),9,10,(11),12,13,(14),15,16,(17),...;
%e A136221 (3),(8),15,(24),34,(46),59,(74),90,108,(127),147,169,(192),...;
%e A136221 (15),(49),108,(198),306,(453),622,(838),1080,1377,(1704),...;
%e A136221 (108),(414),1036,(2116),3493,(5555),8040,(11477),15483,...;
%e A136221 (1036),(4529),12569,(28052),48800,(82328),124335,(186261),...;
%e A136221 (12569),(61369),185704,(446560),811111,(1438447),2250731,...;
%e A136221 ...
%e A136221 and has a recurrence similar to that of square array A136212
%e A136221 which generates the triple factorials.
%o A136221 (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]}
%o A136221 (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]}
%Y A136221 Cf. A136220 (P), A136228 (U), A136231 (W=P^3).
%Y A136221 Cf. other columns of P: A136222, A136223, A136224.
%Y A136221 Cf. related tables: A136217, A136218.
%Y A136221 Cf. variants: A091352, A135881.
%K A136221 nonn
%O A136221 0,3
%A A136221 _Paul D. Hanna_, Dec 25 2007