A177693 - OEIS

%I #11 Apr 09 2024 05:06:26

%S 1,1,1,1,-1,1,1,-1,-1,1,1,3,3,3,1,1,-1,3,3,-1,1,1,-5,-5,15,-5,-5,1,1,

%T 7,35,35,35,35,7,1,1,3,-21,-105,35,-105,-21,3,1,1,-17,51,-357,595,595,

%U -357,51,-17,1,1,11,187,-561,-1309,-6545,-1309,-561,187,11,1

%N Triangle, read by rows, T(n, k) = p(n)/(p(k)*p(n-k)), where p(n) = Product_{j=1..n} A001607(j).

%D Advanced Number Theory, Harvey Cohn, Dover Books, 1963, Page 47ff.

%H G. C. Greubel, <a href="/A177693/b177693.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n, k) = p(n)/(p(k)*p(n-k)), where p(n) = Product_{j=1..n} A001607(j) and p(0) = 1.

%F T(n, n-k) = T(n, k).

%e Triangle begins as:

%e   1;

%e   1,   1;

%e   1,  -1,   1;

%e   1,  -1,  -1,    1;

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

%e   1,  -1,   3,    3,    -1,     1;

%e   1,  -5,  -5,   15,    -5,    -5,     1;

%e   1,   7,  35,   35,    35,    35,     7,    1;

%e   1,   3, -21, -105,    35,  -105,   -21,    3,   1;

%e   1, -17,  51, -357,   595,   595,  -357,   51, -17,  1;

%e   1,  11, 187, -561, -1309, -6545, -1309, -561, 187, 11,  1;

%t A001607:= LinearRecurrence[{-1,-2}, {0,1}, 100];

%t p[n_]:= Product[A001607[[i+1]], {i,n}];

%t T[n_,k_]:= p[n]/(p[k]*p[n-k]);

%t Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

%o (Magma)

%o A001607:=[n le 2 select n-1 else -Self(n-1)-2*Self(n-2): n in [1..100]];

%o p:= func< n | n eq 0 select 1 else (&*[A001607[j+1]: j in [1..n]]) >;

%o A177693:= func< n,k | p(n)/(p(k)*p(n-k)) >;

%o [A177693(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Apr 08 2024

%o (SageMath)

%o A001607=BinaryRecurrenceSequence(-1,-2,0,1)

%o def p(n): return product(A001607(j) for j in range(1,n+1))

%o def A177693(n,k): return p(n)/(p(k)*p(n-k))

%o flatten([[A177693(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Apr 08 2024

%Y Cf. A001607.

%K sign,tabl

%O 0,12

%A _Roger L. Bagula_, May 11 2010

%E Edited by _G. C. Greubel_, Apr 08 2024