cp's OEIS Frontend

A135885 Triangle Q, read by rows, where column k of Q equals column 0 of Q^(k+1) and Q is equal to the matrix square of integer triangle P = A135880 such that column 0 of Q equals column 0 of P shift left.

%I A135885 #3 Mar 30 2012 18:37:07
%S A135885 1,2,1,6,4,1,25,20,6,1,138,126,42,8,1,970,980,351,72,10,1,8390,9186,
%T A135885 3470,748,110,12,1,86796,101492,39968,8936,1365,156,14,1,1049546,
%U A135885 1296934,528306,121532,19090,2250,210,16,1,14563135,18868652,7906598
%N A135885 Triangle Q, read by rows, where column k of Q equals column 0 of Q^(k+1) and Q is equal to the matrix square of integer triangle P = A135880 such that column 0 of Q equals column 0 of P shift left.
%F A135885 See formulas relating triangles P, Q and R, in entry A135880.
%e A135885 Triangle Q = P^2 begins:
%e A135885 1;
%e A135885 2, 1;
%e A135885 6, 4, 1;
%e A135885 25, 20, 6, 1;
%e A135885 138, 126, 42, 8, 1;
%e A135885 970, 980, 351, 72, 10, 1;
%e A135885 8390, 9186, 3470, 748, 110, 12, 1;
%e A135885 86796, 101492, 39968, 8936, 1365, 156, 14, 1;
%e A135885 1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1;
%e A135885 14563135, 18868652, 7906598, 1861416, 298830, 36028, 3451, 272, 18, 1;
%e A135885 228448504, 308478492, 132426050, 31785380, 5193982, 637390, 62230, 5016, 342, 20, 1; ...
%e A135885 where column k of Q equals column 0 of Q^(k+1) for k>=0.
%e A135885 Related triangle P = A135880 begins:
%e A135885 1;
%e A135885 1, 1;
%e A135885 2, 2, 1;
%e A135885 6, 7, 3, 1;
%e A135885 25, 34, 15, 4, 1;
%e A135885 138, 215, 99, 26, 5, 1;
%e A135885 970, 1698, 814, 216, 40, 6, 1; ...
%e A135885 where column k of Q equals column 0 of P^(2k+2)
%e A135885 such that column 0 of P^2 equals column 0 of P shift left.
%e A135885 The matrix product P*R^-1*P = A135899 = Q (shifted down one row),
%e A135885 where R = A135894 begins:
%e A135885 1;
%e A135885 1, 1;
%e A135885 2, 3, 1;
%e A135885 6, 12, 5, 1;
%e A135885 25, 63, 30, 7, 1;
%e A135885 138, 421, 220, 56, 9, 1;
%e A135885 970, 3472, 1945, 525, 90, 11, 1; ...
%e A135885 in which column k of R equals column 0 of P^(2k+1).
%o A135885 (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^2)[n+1,k+1]}
%Y A135885 Cf. columns: A135881, A135886, A135887; related tables: A135880 (P), A135894 (R), A135891 (Q^2), A135893 (Q^3); A135898 (P^-1*R), A135899 (P*R^-1*P), A135900 (R^-1*Q).
%K A135885 nonn,tabl
%O A135885 0,2
%A A135885 _Paul D. Hanna_, Dec 15 2007