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A135893 Triangle, read by rows, equal to P^6, where triangle P = A135880; also equals Q^3 where Q = P^2 = A135885.

%I A135893 #2 Mar 30 2012 18:37:08
%S A135893 1,6,1,42,12,1,351,132,18,1,3470,1554,270,24,1,39968,20260,4089,456,
%T A135893 30,1,528306,294218,65874,8436,690,36,1,7906598,4745522,1147662,
%U A135893 161576,15075,972,42,1,132426050,84534154,21710680,3277148,334390,24486,1302
%N A135893 Triangle, read by rows, equal to P^6, where triangle P = A135880; also equals Q^3 where Q = P^2 = A135885.
%C A135893 Triangle P = A135880 is defined by: column k of P^2 equals column 0 of P^(2k+2) such that column 0 of P^2 equals column 0 of P shift left.
%F A135893 Column k of Q^3 = column 2 of Q^(k+1) for k>=0 where triangle Q = P^2 = A135885; column 0 of Q^3 = column 2 of Q; column 1 of Q^3 = column 2 of Q^2.
%e A135893 Triangle P^6 = Q^3 begins:
%e A135893 1;
%e A135893 6, 1;
%e A135893 42, 12, 1;
%e A135893 351, 132, 18, 1;
%e A135893 3470, 1554, 270, 24, 1;
%e A135893 39968, 20260, 4089, 456, 30, 1;
%e A135893 528306, 294218, 65874, 8436, 690, 36, 1;
%e A135893 7906598, 4745522, 1147662, 161576, 15075, 972, 42, 1;
%e A135893 132426050, 84534154, 21710680, 3277148, 334390, 24486, 1302, 48, 1;
%e A135893 2457643895, 1652665714, 445574768, 70977244, 7732100, 617100, 37149, 1680, 54, 1;
%e A135893 where P = A135880 begins:
%e A135893 1;
%e A135893 1, 1;
%e A135893 2, 2, 1;
%e A135893 6, 7, 3, 1;
%e A135893 25, 34, 15, 4, 1;
%e A135893 138, 215, 99, 26, 5, 1;
%e A135893 970, 1698, 814, 216, 40, 6, 1; ...
%e A135893 and Q = P^2 = A135885 begins:
%e A135893 1;
%e A135893 2, 1;
%e A135893 6, 4, 1;
%e A135893 25, 20, 6, 1;
%e A135893 138, 126, 42, 8, 1;
%e A135893 970, 980, 351, 72, 10, 1;
%e A135893 8390, 9186, 3470, 748, 110, 12, 1; ...
%e A135893 where column k of Q = column 0 of Q^(k+1).
%o A135893 (PARI) {T(n,k)=local(P=Mat(1),R,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]))));R=P*PShR; R=matrix(#P+1, #P+1, r,c, if(r>=c, if(r<#P+1,R[r,c], if(c==1,(P^2)[ #P,1],(P^(2*c-1))[r-c+1,1])))); P=matrix(#R, #R, r,c, if(r>=c, if(r<#R,P[r,c], (R^c)[r-c+1,1])))));(P^6)[n+1,k+1]}
%Y A135893 Cf. A135887 (column 0); A135880 (P), A135885 (Q=P^2), A135891 (Q^2).
%K A135893 nonn,tabl
%O A135893 0,2
%A A135893 _Paul D. Hanna_, Dec 15 2007