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A103371
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Number triangle T(n,k)=C(n,n-k)C(n+1,n-k).
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16
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1, 2, 1, 3, 6, 1, 4, 18, 12, 1, 5, 40, 60, 20, 1, 6, 75, 200, 150, 30, 1, 7, 126, 525, 700, 315, 42, 1, 8, 196, 1176, 2450, 1960, 588, 56, 1, 9, 288, 2352, 7056, 8820, 4704, 1008, 72, 1, 10, 405, 4320, 17640, 31752, 26460, 10080, 1620, 90, 1, 11, 550, 7425, 39600, 97020
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Columns include A000027, A002411, A004302. Row sums are C(2n+1,n+1) or A001700.
T(n-1,k-1) is the number of possibilities to put n identical objects into k of alltogether n distinguishable boxes. See the partition array A035206 from which this triangle arises after summing over all entries related to partitions with fixed part number k.
T(n, k) is also the number of order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). [From A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
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REFERENCES
| 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]
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FORMULA
| Number triangle T(n, k)=C(n, n-k)C(n+1, n-k)=C(n, k)C(n+1, k+1); Column k of this triangle has g.f. sum{j=0..k, C(k, j)C(k+1, j)x^(k+j)}/(1-x)^(2k+2); coefficients of the numerators are the rows of the reverse triangle C(n, k)C(n+1, k).
T(n, k)=C(n, k)*sum{j=0..n, C(n-j, k), j, 0, n-k}; - Paul Barry (pbarry(AT)wit.ie), Jan 12 2006
T(n,k)= (n+1-k)*N(n+1,k+1), with N(n,k):=A001263(n,k), the Narayana triangle (with offset [1,1)]
O.g.f.: ((1-(1-y)*x)/sqrt((1-(1+y)*x)^2-4*x^2*y) -1)/2, (from o.g.f. of A001263, Narayana triangle). W. Lang, Nov 13 2007.
Matrix product of A007318 and A122899. O.g.f. for row n: (1-x)^n*P(n,1,0,(1+x)/(1-x)) = (1-x)^(n-1)*(Legendre_P(n,x) - Legendre_P(n+1,x)), where P(n,a,b,x) denotes the Jacobi polynomial. O.g.f. for column k: x^k/(1-x)^(k+2)*P(k,0,1,(1+x)/(1-x)). Compare with A008459. - Peter Bala (pbala(AT)toucansurf.com), Jan 24 2008
Let S(n,k) = binomial(2*n,n)^(k+1)*((n+1)^(k+1)-n^(k+1))/(n+1)^k. Then T(2*n,n) = S(n,1). (Cf. A194595, A197653, A197654). - Peter Luschny, Oct 20 2011
T(n,k) = A003056(n+1,k+1)C(n,k)^2/(k+1). - Peter Luschny, Oct 29 2011
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EXAMPLE
| Rows start {1}, {2,1}, {3,6,1}, {4,18,12,1},...
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MAPLE
| A103371 := (n, k) -> binomial(n, k)^2*(n+1)/(k+1);
seq(print(seq(A103371(n, k), k=0..n)), n=0..7); # Peter Luschny, Oct 19 2011
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PROG
| (Maxima) create_list(binomial(n, k)*binomial(n+1, k+1), n, 0, 12, k, 0, n); [Emanuele Munarini, Mar 11 2011]
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CROSSREFS
| Cf. A008459, A122899, A194595, A197653.
Sequence in context: A115196 A093346 A115597 * A120257 A059298 A156914
Adjacent sequences: A103368 A103369 A103370 * A103372 A103373 A103374
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Feb 03 2005
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