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A121372
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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.
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1
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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, -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, 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
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OFFSET
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1,23
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COMMENTS
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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))).
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LINKS
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FORMULA
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G.f. of column k: x^(k*(k+1)/2) / ((1+x)(1+x^2)(1+x^3)...(1+x^k)) for k >= 1.
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EXAMPLE
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Triangle begins:
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;
-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, 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, -1;
1, 1, -4, -2, 3, 1;
-1, -1, 4, 2, -3, 0, 1;
1, 0, -4, -2, 4, 2, -1;
-1, 0, 5, 2, -4, -2, 0;
1, 1, -5, -2, 5, 1, -1;
-1, -1, 4, 2, -5, -2, 1;
1, 0, -5, -2, 5, 2, -1;
-1, 0, 6, 3, -6, -3, 1;
1, 1, -5, -3, 6, 2, -1;
-1, -1, 5, 2, -7, -2, 3, 1;
...
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PROG
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(PARI) {T(n, k)=if(n<k|k<1, 0, if(n==1, 1, T(n-k, k-1)-T(n-k, k)))}
(PARI) /* Using generating formula for columns */
{tr(n) = floor((sqrt(1+8*n)-1)/2)} \\ number of terms in row n
{T(n, k) = polcoeff( x^(k*(k+1)/2) / prod(j=1, k, 1 + x^j +x*O(x^n)), n)}
{for(n=1, 50, for(k=1, tr(n), print1(T(n, k), ", ")); print(""))} \\ Paul D. Hanna, Jan 28 2024
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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