A135896 -id:A135896 - cp's OEIS frontend

A135894 Triangle R, read by rows, where column k of R equals column 0 of P^(2k+1) where P=A135880.

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, 8390, 34380, 20340, 5733, 1026, 132, 13, 1, 86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1, 1049546, 5344770, 3430936, 1028076, 194646, 26565, 2808

Offset: 0

Paul D. Hanna, Dec 15 2007

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.

			Triangle R 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;
8390, 34380, 20340, 5733, 1026, 132, 13, 1;
86796, 399463, 247066, 72030, 13305, 1771, 182, 15, 1;
1049546, 5344770, 3430936, 1028076, 194646, 26565, 2808, 240, 17, 1;
14563135, 81097517, 53741404, 16477041, 3182778, 442948, 47801, 4185, 306, 19, 1; ...
where column k of R equals column 0 of P^(2k+1) for k>=0,
and 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; ...
where column k of P equals column 0 of R^(k+1).
The matrix product P^-1*R = A135898 = P (shifted right one column);
the matrix product R^-1*P^2 = A135900 = R (shifted down one row).

Cf. A135881 (column 0), A135889 (column 1); A135880 (P), A135885 (Q=P^2), A135895 (R^2), A135896 (R^3), A135897 (R^4); A135888 (P^3) A135892 (P^5); A135898 (P^-1*R), A135899 (P*R^-1*P), A135900 (R^-1*Q).

{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])))));R[n+1,k+1]}

Column k of R = column 0 of P^(2k+1) for k>=0 where triangle P = A135880; column 0 of R = column 0 of P; column 1 of R = column 0 of P^3; column 2 of R = column 0 of P^5. See more formulas relating triangles P, Q and R, in entry A135880.

Typo in formula corrected by Paul D. Hanna, Mar 26 2010

A135892 Triangle, read by rows, equal to P^5, where triangle P = A135880.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Dec 15 2007

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.

			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).

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

{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]}

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.

cp's OEIS Frontend

A135894 Triangle R, read by rows, where column k of R equals column 0 of P^(2k+1) where P=A135880.

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