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A135899 Triangle, read by rows equal to the matrix product P*R^-1*P, where P = A135880 and R = A135894; P*R^-1*P equals triangle Q=A135885 shifted down one row. 5
1, 1, 1, 2, 1, 1, 6, 4, 1, 1, 25, 20, 6, 1, 1, 138, 126, 42, 8, 1, 1, 970, 980, 351, 72, 10, 1, 1, 8390, 9186, 3470, 748, 110, 12, 1, 1, 86796, 101492, 39968, 8936, 1365, 156, 14, 1, 1, 1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1, 1, 14563135, 18868652 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..56.

EXAMPLE

Triangle begins:

1;

1, 1;

2, 1, 1;

6, 4, 1, 1;

25, 20, 6, 1, 1;

138, 126, 42, 8, 1, 1;

970, 980, 351, 72, 10, 1, 1;

8390, 9186, 3470, 748, 110, 12, 1, 1;

86796, 101492, 39968, 8936, 1365, 156, 14, 1, 1;

1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1, 1; ...

This triangle equals matrix product P*R^-1*P,

which equals triangle Q shifted down one row,

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; ...

and 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; ...

where column k of R equals column 0 of P^(2k+1),

and column k of Q=P^2 equals column 0 of P^(2k+2), for k>=0.

PROG

(PARI) {T(n, k)=local(P=Mat(1), R=Mat(1), 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*R^-1*P)[n+1, k+1]}

CROSSREFS

Cf. A135880 (P), A135885 (Q=P^2), A135894 (R); A135898 (P^-1*R), A135900 (R^-1*Q).

Sequence in context: A257895 A186023 A103880 * A327816 A047920 A249673

Adjacent sequences:  A135896 A135897 A135898 * A135900 A135901 A135902

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 15 2007

STATUS

approved

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Last modified August 12 21:10 EDT 2020. Contains 336440 sequences. (Running on oeis4.)