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Search: id:A121435
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| A121435 |
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Matrix inverse of triangle A122175, where A122175(n,k) = C( k*(k+1)/2 + n-k, n-k) for n>=k>=0. |
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+0
4
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| 1, -1, 1, 1, -2, 1, -2, 5, -4, 1, 7, -19, 18, -7, 1, -37, 104, -106, 49, -11, 1, 268, -766, 809, -406, 110, -16, 1, -2496, 7197, -7746, 4060, -1210, 216, -22, 1, 28612, -82910, 90199, -48461, 15235, -3032, 385, -29, 1, -391189, 1136923, -1244891, 678874, -220352, 46732, -6699, 638, -37, 1
(list; table; graph; listen)
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OFFSET
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0,5
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FORMULA
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(1) T(n,k) = A121434(n-1,k) - A121434(n-1,k+1). (2) T(n,k) = (-1)^(n-k)*[A107876^(k*(k+1)/2 + 1)](n,k); i.e., column k equals signed column k of A107876^(k*(k+1)/2 + 1). 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 + 1); (4) 1 = Sum_{j>=0} T(j+k,k)*x^j*(1+x)^( j*(j-1)/2) + j*k + k*(k+1)/2 + 1).
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EXAMPLE
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Triangle begins:
1;
-1, 1;
1, -2, 1;
-2, 5, -4, 1;
7, -19, 18, -7, 1;
-37, 104, -106, 49, -11, 1;
268, -766, 809, -406, 110, -16, 1;
-2496, 7197, -7746, 4060, -1210, 216, -22, 1;
28612, -82910, 90199, -48461, 15235, -3032, 385, -29, 1;
-391189, 1136923, -1244891, 678874, -220352, 46732, -6699, 638, -37, 1; ...
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PROGRAM
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(PARI) /* Matrix Inverse of A122175 */ {T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, binomial((c-1)*(c-2)/2+r-1, 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+1)), n-k)}
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CROSSREFS
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Cf. A098568, A107876; unsigned columns: A107877, A107882.
Sequence in context: A151703 A151691 A104560 this_sequence A137156 A136457 A078016
Adjacent sequences: A121432 A121433 A121434 this_sequence A121436 A121437 A121438
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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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