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%I #16 Jan 29 2024 01:18:42
%S 1,-1,1,1,-1,-1,1,0,-1,0,1,1,1,-1,-1,-1,0,1,0,-1,-1,0,2,1,1,1,-1,-1,
%T -1,-1,1,0,1,0,-2,-1,-1,0,2,1,1,1,-2,0,1,-1,-1,2,1,-1,1,0,-2,-1,0,-1,
%U 0,3,1,-1,1,1,-3,-2,1,-1,-1,2,1,-1,1,0,-3,-1,2,1,-1,0,4,2,-1,-1,1,1,-3,-1,2,0,-1,-1,3,1,-3,-1,1,0,-4,-2,2,1,-1,0,4,2,-3
%N Triangle, read by rows of length A003056(n) for n >= 1, defined by the recurrence: T(n,k) = T(n-k,k-1) - T(n-k,k) for n > k > 1, with T(n,1) =(-1)^(n-1) for n >= 1.
%C Row sums equal A003406 (offset 1), the expansion of Ramanujan's function: R(x) = 1 + Sum_{n>=1} (x^(n*(n+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^n))).
%H Paul D. Hanna, <a href="/A121372/b121372.txt">Table of n, a(n) for n = 1..10075</a>
%F G.f. of column k: x^(k*(k+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^k)) for k >= 1.
%e Triangle begins:
%e 1;
%e -1;
%e 1, 1;
%e -1, -1;
%e 1, 0;
%e -1, 0, 1;
%e 1, 1, -1;
%e -1, -1, 0;
%e 1, 0, -1;
%e -1, 0, 2, 1;
%e 1, 1, -1, -1;
%e -1, -1, 1, 0;
%e 1, 0, -2, -1;
%e -1, 0, 2, 1;
%e 1, 1, -2, 0, 1;
%e -1, -1, 2, 1, -1;
%e 1, 0, -2, -1, 0;
%e -1, 0, 3, 1, -1;
%e 1, 1, -3, -2, 1;
%e -1, -1, 2, 1, -1;
%e 1, 0, -3, -1, 2, 1;
%e -1, 0, 4, 2, -1, -1;
%e 1, 1, -3, -1, 2, 0;
%e -1, -1, 3, 1, -3, -1;
%e 1, 0, -4, -2, 2, 1;
%e -1, 0, 4, 2, -3, -1;
%e 1, 1, -4, -2, 3, 1;
%e -1, -1, 4, 2, -3, 0, 1;
%e 1, 0, -4, -2, 4, 2, -1;
%e -1, 0, 5, 2, -4, -2, 0;
%e 1, 1, -5, -2, 5, 1, -1;
%e -1, -1, 4, 2, -5, -2, 1;
%e 1, 0, -5, -2, 5, 2, -1;
%e -1, 0, 6, 3, -6, -3, 1;
%e 1, 1, -5, -3, 6, 2, -1;
%e -1, -1, 5, 2, -7, -2, 3, 1;
%e ...
%o (PARI) {T(n, k)=if(n<k|k<1, 0, if(n==1, 1, T(n-k, k-1)-T(n-k, k)))}
%o (PARI) /* Using generating formula for columns */
%o {tr(n) = floor((sqrt(1+8*n)-1)/2)} \\ number of terms in row n
%o {T(n,k) = polcoeff( x^(k*(k+1)/2) / prod(j=1,k, 1 + x^j +x*O(x^n)), n)}
%o {for(n=1,50, for(k=1, tr(n), print1(T(n,k),", "));print(""))} \\ _Paul D. Hanna_, Jan 28 2024
%Y Cf. A003406, A008289.
%K sign,tabf
%O 1,23
%A _Paul D. Hanna_, Jul 24 2006
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