A132845 - OEIS

%I #7 Mar 14 2015 12:05:43

%S 1,1,1,3,2,1,13,9,3,1,73,42,18,4,1,466,270,95,30,5,1,3309,1785,693,

%T 179,45,6,1,25425,13657,4893,1463,301,63,7,1,209717,108606,40506,

%U 11104,2726,468,84,8,1,1837168,943677,338277,99177,22239,4653,687,108,9,1

%N Triangle, read by rows, where row n equals row n of matrix power A132844^n for n>=0, where triangle A132844 is defined by: A132844(n,k) = T( [(n+k)/2], k) for n>=k>=0.

%F T(n,k) = [A132844^n](n,k) where A132844(n,k) = T( [(n+k)/2], k) for n>=k>=0.

%e Triangle T begins:

%e 1;

%e 1, 1;

%e 3, 2, 1;

%e 13, 9, 3, 1;

%e 73, 42, 18, 4, 1;

%e 466, 270, 95, 30, 5, 1;

%e 3309, 1785, 693, 179, 45, 6, 1;

%e 25425, 13657, 4893, 1463, 301, 63, 7, 1;

%e 209717, 108606, 40506, 11104, 2726, 468, 84, 8, 1;

%e 1837168, 943677, 338277, 99177, 22239, 4653, 687, 108, 9, 1;

%e 16995545, 8534290, 3110310, 873440, 213415, 40707, 7440, 965, 135, 10, 1; ...

%e Triangle A132844 begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 2, 1, 1;

%e 3, 2, 3, 1, 1;

%e 3, 9, 3, 4, 1, 1;

%e 13, 9, 18, 4, 5, 1, 1; ...

%e where column k of A132844 equals column k of T with terms repeated;

%e then row n of T equals row n of A132844^n as illustrated below.

%e Matrix square A132844^2 begins:

%e 1;

%e 2, 1;

%e 3, 2, 1; <-- row 2 of T

%e 5, 5, 2, 1;

%e 12, 9, 7, 2, 1;

%e 25, 31, 13, 9, 2, 1; ...

%e Matrix cube A132844^3 begins:

%e 1;

%e 3, 1;

%e 6, 3, 1;

%e 13, 9, 3, 1; <-- row 3 of T

%e 33, 22, 12, 3, 1;

%e 87, 75, 31, 15, 3, 1; ...

%e Matrix 4th power A132844^4 begins:

%e 1;

%e 4, 1;

%e 10, 4, 1;

%e 26, 14, 4, 1;

%e 73, 42, 18, 4, 1; <-- row 4 of T

%e 220, 151, 58, 22, 4, 1; ...

%t t[n_, k_] := t[n, k] = Module[{m, p}, m = Table[ Which[c < r, t[Quotient[r+c, 2]-1, c-1], c == r, 1, True, 0], {r, 1, n+1}, {c, 1, n+1}]; p = MatrixPower[m, n]; If[k > n, 0, p[[n+1, k+1]]]]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 02 2013, after Pari *)

%o (PARI) {T(n,k)=local(M=matrix(n+1,n+1,r,c,if(r>=c,if(r<=c+1,1,T((r+c)\2-1,c-1))))); (M^n)[n+1,k+1]}

%Y Cf. A132844 (triangle); columns: A132846, A132847, A132848, A132849.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Sep 17 2007