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.

A135891 Triangle, read by rows, equal to P^4, where triangle P = A135880; also equals Q^2, where triangle Q = P^2 = A135885.

Original entry on oeis.org

1, 4, 1, 20, 8, 1, 126, 64, 12, 1, 980, 580, 132, 16, 1, 9186, 6064, 1554, 224, 20, 1, 101492, 72832, 20260, 3240, 340, 24, 1, 1296934, 995050, 294218, 50496, 5830, 480, 28, 1, 18868652, 15301004, 4745522, 857840, 105620, 9516, 644, 32, 1, 308478492
Offset: 0

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Author

Paul D. Hanna, Dec 15 2007

Keywords

Examples

			Triangle P^4 = Q^2 begins:
1;
4, 1;
20, 8, 1;
126, 64, 12, 1;
980, 580, 132, 16, 1;
9186, 6064, 1554, 224, 20, 1;
101492, 72832, 20260, 3240, 340, 24, 1;
1296934, 995050, 294218, 50496, 5830, 480, 28, 1;
18868652, 15301004, 4745522, 857840, 105620, 9516, 644, 32, 1;
308478492, 262203558, 84534154, 15907004, 2052450, 196400, 14490, 832, 36, 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 equals column 0 of Q^(k+1) for k>=0
and column 0 of Q equals column 0 of P shift left.
		

Crossrefs

Cf. A135886 (column 0); A135880 (P), A135885 (Q=P^2), A135893 (Q^3).

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^4)[n+1,k+1]}

Formula

Column k of Q^2 = column 1 of Q^(k+1) for k>=0 where triangle Q = P^2 = A135885. 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.