A135891 Triangle, read by rows, equal to P^4, where triangle P = A135880; also equals Q^2, where triangle Q = P^2 = A135885.

1, 4, 1, 20, 8, 1, 126, 64, 12, 1, 980, 580, 132, 16, 1, 9186, 6064, 1554, 224, 20, 1, 101492, 72832, 20260, 3240, 340, 24, 1, 1296934, 995050, 294218, 50496, 5830, 480, 28, 1, 18868652, 15301004, 4745522, 857840, 105620, 9516, 644, 32, 1, 308478492

Offset

0,2

Links

Table of n, a(n) for n=0..45.

Formula

Column k of Q^2 = column 1 of Q^(k+1) for k>=0 where triangle Q = P^2 = A135885. 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.

Example

Triangle P^4 = Q^2 begins:
1;
4, 1;
20, 8, 1;
126, 64, 12, 1;
980, 580, 132, 16, 1;
9186, 6064, 1554, 224, 20, 1;
101492, 72832, 20260, 3240, 340, 24, 1;
1296934, 995050, 294218, 50496, 5830, 480, 28, 1;
18868652, 15301004, 4745522, 857840, 105620, 9516, 644, 32, 1;
308478492, 262203558, 84534154, 15907004, 2052450, 196400, 14490, 832, 36, 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; ...
and Q = P^2 = A135885 begins:
1;
2, 1;
6, 4, 1;
25, 20, 6, 1;
138, 126, 42, 8, 1;
970, 980, 351, 72, 10, 1;
8390, 9186, 3470, 748, 110, 12, 1; ...
where column k of Q equals column 0 of Q^(k+1) for k>=0
and column 0 of Q equals column 0 of P shift left.

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^4)[n+1, k+1]}

Crossrefs

Cf. A135886 (column 0); A135880 (P), A135885 (Q=P^2), A135893 (Q^3).

Sequence in context: A167432 A201639 A078939 * A049459 A143493 A062137

Adjacent sequences: A135888 A135889 A135890 * A135892 A135893 A135894

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 15 2007

Status

approved