OFFSET
0,2
COMMENTS
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.
FORMULA
Column k of P^5 = column 2 of R^(k+1) for k>=0 where triangle R = A135894; column 0 of P^5 = column 2 of R; column 1 of P^5 = column 2 of R^2; column 2 of P^5 = column 2 of R^3; column 3 of P^5 = column 2 of R^4.
EXAMPLE
Triangle P^5 begins:
1;
5, 1;
30, 10, 1;
220, 95, 15, 1;
1945, 990, 195, 20, 1;
20340, 11635, 2625, 330, 25, 1;
247066, 154450, 38270, 5440, 500, 30, 1;
3430936, 2302142, 611525, 94515, 9750, 705, 35, 1;
53741404, 38229214, 10721093, 1761940, 196500, 15870, 945, 40, 1;
938816814, 701685738, 205607124, 35429974, 4182295, 363820, 24115, 1220, 45, 1;
where P = A135880 begins:
1;
1, 1;
2, 2, 1;
6, 7, 3, 1;
25, 34, 15, 4, 1;
138, 215, 99, 26, 5, 1;
970, 1698, 814, 216, 40, 6, 1; ...
in which column k of P = column 0 of R^(k+1),
where R = A135894 begins:
1;
1, 1;
2, 3, 1;
6, 12, 5, 1;
25, 63, 30, 7, 1;
138, 421, 220, 56, 9, 1;
970, 3472, 1945, 525, 90, 11, 1; ...
in which column k of R equals column 0 of P^(2k+1).
PROG
(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^5)[n+1, k+1]}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Dec 15 2007
STATUS
approved