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Search: id:A054142
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| A054142 |
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Triangular array C(2n-k,k), k=0,1,...,n, n >= 0. |
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+0
13
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| 1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 15, 10, 1, 1, 9, 28, 35, 15, 1, 1, 11, 45, 84, 70, 21, 1, 1, 13, 66, 165, 210, 126, 28, 1, 1, 15, 91, 286, 495, 462, 210, 36, 1, 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 1, 21, 190
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row sums are odd-indexed Fibonacci numbers.
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k double rises. Mirror image of A085478. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
Diagonal sums are A052535. - Paul Barry (pbarry(AT)wit.ie), Jan 21 2005
Matrix inverse is the triangle of Salie numbers A098435. - Paul Barry (pbarry(AT)wit.ie), Jan 21 2005
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REFERENCES
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E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
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LINKS
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D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
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FORMULA
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G.f.=(1-tz)/[(1-tz)^2-z]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
Column k has g.f. sum{j=0..k+1, binomial(k+1, 2j)x^j}*x^k/(1-x)^(k+1); - Paul Barry (pbarry(AT)wit.ie), Jun 22 2005
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EXAMPLE
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Rows: {1}, {1,1}, {1,3,1}, {1,5,6,1}, ...
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MAPLE
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T:=(n, k)->binomial(2*n-k, k): seq(seq(T(n, k), k=0..n), n=0..11);
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PROGRAM
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(PARI) T(n, k)=if(n<0, 0, polcoeff(charpoly(matrix(n, n, i, j, -min(i, j))), k))
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CROSSREFS
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These are the even-indexed rows of A011973, the odd-indexed rows form A053123.
Cf. A076756, A085478.
Sequence in context: A073145 A145033 A076756 this_sequence A114172 A121522 A080842
Adjacent sequences: A054139 A054140 A054141 this_sequence A054143 A054144 A054145
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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