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A085478
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Triangle read by rows: T(n, k) = binomial(n + k, 2*k).
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39
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1, 1, 1, 1, 3, 1, 1, 6, 5, 1, 1, 10, 15, 7, 1, 1, 15, 35, 28, 9, 1, 1, 21, 70, 84, 45, 11, 1, 1, 28, 126, 210, 165, 66, 13, 1, 1, 36, 210, 462, 495, 286, 91, 15, 1, 1, 45, 330, 924, 1287, 1001, 455, 120, 17, 1, 1, 55, 495, 1716, 3003, 3003, 1820, 680, 153, 19, 1, 1, 66, 715
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Coefficient array for Morgan-Voyce polynomial b(n,x). A053122 (unsigned) is the coefficient array for B(n,x). Reversal of A054142. - Paul Barry (pbarry(AT)wit.ie), Jan 19 2004
This triangle is formed from even-numbered rows of triangle A011973 read in reverse order. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k+1 peaks. T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k peaks at height >=2. T(n,k) is the number of directed column-convex polyominoes of area n+1, having k+1 columns. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
Riordan array (1/(1-x),x/(1-x)^2). - Paul Barry (pbarry(AT)wit.ie), May 09 2005
The triangular matrix a(n,k) = (-1)^(n+k)*T(n,k) is the matrix inverse of A039599 . - Philippe DELEHAM, May 26 2005
The n-th row gives absolute values of coefficients of reciprocal of g.f. of bottom-line of n-wave sequence. - Floor van Lamoen (fvlamoen(AT)planet.nl), Sep 24 2006
Unsigned version of A129818 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 25 2007
T(n, k) is also the number of idempotent order-preserving full transformations (of an n-chain) of height k >=1 (height(alpha) = |Im(alpha)|) and of waist n (waist(alpha) = max(Im(alpha))). [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
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REFERENCES
| E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
E. Deutsch and H. Prodinger, A bijection between directed column-convex polyominoes and ordered trees of height at most three, Theoretical Comp. Science, 307, 2003, 319-325.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving full transformations. Semigroup Forum 72 (2006), 51-62. [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370.
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LINKS
| Eric Weisstein's World of Mathematics, Morgan-Voyce Polynomials
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FORMULA
| T(n, k) = (n+k)!/((n-k)!*(2*k)!)
G.f.=(1-z)/[(1-z)^2-tz]. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
Row sums are A001519 (Fib(2n+1)). Diagonal sums are A011782. Binomial transform of A026729 (product of lower triangular matrices). - Paul Barry (pbarry(AT)wit.ie), Jun 21 2004
T(n, 0) = 1, T(n, k) = 0 if n<k; T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) . T(0, 0) = 1, T(0, k) = 0 if k>0; T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) . For the column k, g.f.: Sum_{n>=0} T(n, k)*x^n = (x^k) / (1-x)^(2*k+1) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 15 2004
Sum_{k, 0<=k<=n}T(n,k)*x^(2*k) = A000012(n), A001519(n+1), A001653(n), A078922(n+1), A007805(n), A097835(n), A097315(n), A097838(n), A078988(n), A097841(n), A097727(n), A097843(n), A097730(n), A098244(n), A097733(n), A098247(n), A097736(n), A098250(n), A097739(n), A098253(n), A097742(n), A098256(n), A097767(n), A098259(n), A097770(n), A098262(n), A097773(n), A098292(n), A097776(n) for x=0,1,2,...27,28 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 31 2007
T(2n,n)=A005809(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009]
A183160(n) = Sum_{k=0..n} T(n,k)*T(n,n-k). [Paul D. Hanna (pauldhanna(AT)juno.com), Dec 27 2010]
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - DELEHAM Philippe, Feb 06 2012
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EXAMPLE
| 1
1 1
1 3 1
1 6 5 1
1 10 15 7 1
1 15 35 28 9 1
1 21 70 84 45 11 1
1 28 126 210 165 66 13 1
1 36 210 462 495 286 91 15 1
1 45 330 924 1287 1001 455 120 17 1
1 55 495 1716 3003 3003 1820 680 153 19 1
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MAPLE
| T:=(n, k)->binomial(n+k, 2*k): seq(seq(T(n, k), k=0..n), n=0..11);
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PROG
| (PARI) T(n, k)=binomial(n+k, n-k)
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CROSSREFS
| Row sums: A001519. Cf. A007318.
Cf. A098158 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
Cf. A183160.
Sequence in context: A121524 A103141 A129818 * A123970 A055898 A145904
Adjacent sequences: A085475 A085476 A085477 * A085479 A085480 A085481
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KEYWORD
| easy,nonn,tabl,changed
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AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Aug 14 2003
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