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

1, 6, 1, 42, 12, 1, 351, 132, 18, 1, 3470, 1554, 270, 24, 1, 39968, 20260, 4089, 456, 30, 1, 528306, 294218, 65874, 8436, 690, 36, 1, 7906598, 4745522, 1147662, 161576, 15075, 972, 42, 1, 132426050, 84534154, 21710680, 3277148, 334390, 24486, 1302

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.

Links

Table of n, a(n) for n=0..42.

Formula

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.

Example

Triangle P^6 = Q^3 begins:
1;
6, 1;
42, 12, 1;
351, 132, 18, 1;
3470, 1554, 270, 24, 1;
39968, 20260, 4089, 456, 30, 1;
528306, 294218, 65874, 8436, 690, 36, 1;
7906598, 4745522, 1147662, 161576, 15075, 972, 42, 1;
132426050, 84534154, 21710680, 3277148, 334390, 24486, 1302, 48, 1;
2457643895, 1652665714, 445574768, 70977244, 7732100, 617100, 37149, 1680, 54, 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 = column 0 of Q^(k+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^6)[n+1, k+1]}

Crossrefs

Cf. A135887 (column 0); A135880 (P), A135885 (Q=P^2), A135891 (Q^2).

Sequence in context: A145356 A145357 A035529 * A051338 A062138 A143498

Adjacent sequences: A135890 A135891 A135892 * A135894 A135895 A135896

Keyword

nonn,tabl

Author

Paul D. Hanna, Dec 15 2007

Status

approved