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A078812
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Triangle read by rows: T(n,k) = binomial(n+k-1,2*k-1).
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20
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1, 2, 1, 3, 4, 1, 4, 10, 6, 1, 5, 20, 21, 8, 1, 6, 35, 56, 36, 10, 1, 7, 56, 126, 120, 55, 12, 1, 8, 84, 252, 330, 220, 78, 14, 1, 9, 120, 462, 792, 715, 364, 105, 16, 1, 10, 165, 792, 1716, 2002, 1365, 560, 136, 18, 1, 11, 220, 1287, 3432, 5005, 4368, 2380, 816, 171, 20
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Apart from signs, identical to A053122.
Coefficient array for Morgan-Voyce polynomial B(n,x); see A085478 for references. DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004
T(n,k)=number of compositions of n having k parts when there are q kinds of part q (q=1,2,...). Example: T(4,2)=10 because we have (1,3),(1,3'),(1,3"), (3,1),(3',1),(3",1),(2,2),(2,2'),(2',2) and (2',2'). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005
T(n, k) is also the number of idempotent 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]
Yidong Sun, Numerical triangles and several classical sequences, Fib. Quart., Nov. 2005, pp. 359-370.
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LINKS
| T. D. Noe, Rows n=0..50 of triangle, flattened
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FORMULA
| G.f.: x*y/(1-(2+y)*x+x^2). To get row n, expand this in powers of x then expand the coefficient of x^n in increasing powers of y.
If indexing begins at 0 we have: T(n, k) = (n+k+1)!/((n-k)!*(2k+1))!. T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*(j+1) with T(n, 0) = n+1, T(n, k) = 0 if n<k. T(n, k) = T(n-1, k-1) + T(n-1, k) + Sum_{j>=0} (-1)^j*T(n-1, k+j)*A000108(j) with T(n, k) = 0 if k<0, T(0, 0)=1 and T(0, k) = 0 for k>0. G.f. for the column k : Sum_{n>=0} T(n, k)*x^n = (x^k)/(1-x)^(2k+2). Row sums : Sum_{k>=0} T(n, k) = A001906(n+1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 16 2004
Diagonal sums are A000079(n)=sum{k=0..floor(n/2), binomial(n+k+1, n-k)}. - Paul Barry (pbarry(AT)wit.ie), Jun 21 2004
Riordan array (1/(1-x)^2, x/(1-x)^2). - Paul Barry (pbarry(AT)wit.ie), Oct 22 2006
T(0,0)= 1, T(n,k)=0 if k<0 or if k>n, T(n,k) = T(n-1,k-1)+2*T(n-1,k)-T(n-2,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 26 2010]
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EXAMPLE
| Triangle begins:
.........................1
........................2,.1
......................3,.4,.1
....................4,.10,.6,.1
..................5,.20,.21,.8,.1
................6,.35,.56,.36,.10,.1
.............7,.56,.126,.120,.55,.12,.1
..........8,.84,.252,.330,.220,.78,.14,.1
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MAPLE
| for n from 1 to 11 do seq(binomial(n+k-1, 2*k-1), k=1..n) od; # yields sequence in triangular form (Deutsch)
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PROG
| (PARI) T(n, k)=if(n<0, 0, binomial(n+k-1, 2*k-1))
(PARI) {T(n, k)=polcoeff( polcoeff( x*y/(1-(2+y)*x+x^2) +x*O(x^n), n), k)}
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CROSSREFS
| This triangle is formed from odd-numbered rows of triangle A011973 read in reverse order.
The column sequences are A000027, A000292, A000389, A000580, A000582, A001288 for k=1..6, resp. For k=7..24 they are A010966..(+2)..A011000 and for k=25..50 they are A017713..(+2)..A017763.
Cf. A053123, A049310. Row sums give A001906.
With signs: A053122.
Cf. A119900 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
Sequence in context: A143326 A186686 A053122 * A104711 A133112 A198060
Adjacent sequences: A078809 A078810 A078811 * A078813 A078814 A078815
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KEYWORD
| easy,nice,nonn,tabl
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AUTHOR
| Michael Somos, Dec 05, 2002
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Apr 28 2008
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