cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A135885 Triangle Q, read by rows, where column k of Q equals column 0 of Q^(k+1) and Q is equal to the matrix square of integer triangle P = A135880 such that column 0 of Q equals column 0 of P shift left.

Original entry on oeis.org

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, 86796, 101492, 39968, 8936, 1365, 156, 14, 1, 1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1, 14563135, 18868652, 7906598
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Examples

			Triangle Q = P^2 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;
86796, 101492, 39968, 8936, 1365, 156, 14, 1;
1049546, 1296934, 528306, 121532, 19090, 2250, 210, 16, 1;
14563135, 18868652, 7906598, 1861416, 298830, 36028, 3451, 272, 18, 1;
228448504, 308478492, 132426050, 31785380, 5193982, 637390, 62230, 5016, 342, 20, 1; ...
where column k of Q equals column 0 of Q^(k+1) for k>=0.
Related triangle 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 Q equals column 0 of P^(2k+2)
such that column 0 of P^2 equals column 0 of P shift left.
The matrix product P*R^-1*P = A135899 = Q (shifted down one row),
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).

Crossrefs

Cf. columns: A135881, A135886, A135887; related tables: A135880 (P), A135894 (R), A135891 (Q^2), A135893 (Q^3); A135898 (P^-1*R), A135899 (P*R^-1*P), A135900 (R^-1*Q).

Programs

  • 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^2)[n+1,k+1]}

Formula

See formulas relating triangles P, Q and R, in entry A135880.

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

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

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.

Examples

			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).

Crossrefs

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

Programs

  • 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]}

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.

A135886 Column 1 of triangle Q = A135885; also equals column 0 of Q^2.

Original entry on oeis.org

1, 4, 20, 126, 980, 9186, 101492, 1296934, 18868652, 308478492, 5605768476, 112198139500, 2454071216496, 58267971181456, 1493114371576942, 41084194594171729, 1208473333806735096, 37849717704435895370
Offset: 0

Views

Author

Paul D. Hanna, Dec 15 2007

Keywords

Examples

			Triangle Q = 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) such that
column 0 of Q equals column 0 of P=A135880 shift left and Q=P^2.

Crossrefs

Cf. A135885; other columns: A135881, A135887.

Programs

  • PARI
    {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^2)[n+2,2])}
Showing 1-3 of 3 results.

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