_Paul D. Hanna (pauldhanna(AT)juno.com), _, Dec 15 2007

Triangle, read by rows, equal to P^5, where triangle P = A135880.

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

0,2

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.

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.

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).

(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]}

Cf. A135880 (P), A135894 (R), A135895 (R^2), A135896 (R^3), A135897 (R^4).

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 15 2007

approved