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Search: id:A121436
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| A121436 |
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Matrix inverse of triangle A122176, where A122176(n,k) = C( k*(k+1)/2 + n-k + 1, n-k) for n>=k>=0. |
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+0
4
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| 1, -2, 1, 3, -3, 1, -7, 9, -5, 1, 26, -37, 25, -8, 1, -141, 210, -155, 60, -12, 1, 1034, -1575, 1215, -516, 126, -17, 1, -9693, 14943, -11806, 5270, -1426, 238, -23, 1, 111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1, -1528112, 2381814, -1923765, 899226, -267084, 53431, -7337, 675, -38, 1
(list; table; graph; listen)
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OFFSET
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0,2
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FORMULA
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(1) T(n,k) = A121435(n-1,k) - A121435(n-1,k+1). (2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 2)](n,k); i.e., column k equals signed column k of A107876^(k*(k+1)/2 + 2). G.f.s for column k: (3) 1 = Sum_{j>=0} T(j+k,k)*x^j/(1-x)^( j*(j+1)/2) + j*k + k*(k+1)/2 + 2); (4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 2).
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EXAMPLE
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Triangle begins:
1;
-2, 1;
3, -3, 1;
-7, 9, -5, 1;
26, -37, 25, -8, 1;
-141, 210, -155, 60, -12, 1;
1034, -1575, 1215, -516, 126, -17, 1;
-9693, 14943, -11806, 5270, -1426, 238, -23, 1;
111522, -173109, 138660, -63696, 18267, -3417, 414, -30, 1;
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PROGRAM
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(PARI) /* Matrix Inverse of A122176 */ {T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, binomial((c-1)*(c-2)/2+r, r-c)))); return((M^-1)[n+1, k+1])} (PARI) /* Obtain by G.F. */ {T(n, k)=polcoeff(1-sum(j=0, n-k-1, T(j+k, k)*x^j/(1-x+x*O(x^n))^(j*(j+1)/2+j*k+k*(k+1)/2+2)), n-k)}
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CROSSREFS
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Cf. A098568, A107876; unsigned columns: A107881, A107886.
Sequence in context: A165007 A127123 A103525 this_sequence A088074 A071463 A047679
Adjacent sequences: A121433 A121434 A121435 this_sequence A121437 A121438 A121439
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KEYWORD
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sign,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Aug 27 2006
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