A135894
Triangle R, read by rows, where column k of R equals column 0 of P^(2k+1) where P=A135880.
Original entry on oeis.org
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
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).
-
{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]}
A135888
Triangle, read by rows, equal to the matrix cube of triangle P = A135880.
Original entry on oeis.org
1, 3, 1, 12, 6, 1, 63, 39, 9, 1, 421, 300, 81, 12, 1, 3472, 2741, 816, 138, 15, 1, 34380, 29380, 9366, 1716, 210, 18, 1, 399463, 363922, 122148, 23647, 3105, 297, 21, 1, 5344770, 5135894, 1795481, 362116, 49880, 5088, 399, 24, 1, 81097517, 81557270
Offset: 0
Triangle P^3 begins:
1;
3, 1;
12, 6, 1;
63, 39, 9, 1;
421, 300, 81, 12, 1;
3472, 2741, 816, 138, 15, 1;
34380, 29380, 9366, 1716, 210, 18, 1;
399463, 363922, 122148, 23647, 3105, 297, 21, 1;
5344770, 5135894, 1795481, 362116, 49880, 5088, 399, 24, 1;
81097517, 81557270, 29478724, 6138746, 875935, 93306, 7770, 516, 27, 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; ...
where 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.
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{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^3)[n+1,k+1]}
A135890
Column 1 of triangle A135888, which equals the matrix cube of triangle A135880; also equals column 1 of triangle A135895.
Original entry on oeis.org
1, 6, 39, 300, 2741, 29380, 363922, 5135894, 81557270, 1441771540, 28114817877, 600012111858, 13919315033624, 348932593149877, 9403371859278914, 271183690566871863, 8335374900994682248, 272083236017290793444
Offset: 0
-
{a(n)=local(P=Mat(1),R,PShR);if(n==0,1,for(i=0,n+1, 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^3)[n+2,2])}
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