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 A103371 Number triangle T(n,k) = C(n,n-k)*C(n+1,n-k). 21

%I

%S 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,

%T 315,42,1,8,196,1176,2450,1960,588,56,1,9,288,2352,7056,8820,4704,

%U 1008,72,1,10,405,4320,17640,31752,26460,10080,1620,90,1,11,550,7425,39600,97020

%N Number triangle T(n,k) = C(n,n-k)*C(n+1,n-k).

%C Columns include A000027, A002411, A004302, A108647, A134287. Row sums are C(2n+1,n+1) or A001700.

%C T(n-1,k-1) is the number of ways to put n identical objects into k of altogether 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.

%C T(n, k) is also the number of order-preserving full transformations (of an n-chain) of height k (height(alpha) = |Im(alpha)|). - _Abdullahi Umar_, Oct 02 2008

%C From _R. J. Mathar_, Mar 29 2013: (Start)

%C The matrix inverse starts

%C 1;

%C -2, 1;

%C 9, -6, 1;

%C -76, 54, -12, 1;

%C 1055, -760, 180, -20, 1;

%C -21906, 15825, -3800, 450, -30, 1;

%C 636447, -460026, 110775, -13300, 945, -42, 1; (End)

%C The o.g.f. of the (n+1)-th diagonal is given by G(n, x) = (n+1)*Sum_{k=1..n} A001263(n, k)*x^(k-1) / (1 - x)^(2*n+1), for n >= 1 and for n = 0 it is G(0, x) = 1/(1-x). - _Wolfdieter Lang_, Jul 31 2017

%H Reinhard Zumkeller, <a href="/A103371/b103371.txt">Rows n = 0..125 of table, flattened</a>

%H P. Barry, A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry2/barry126.html">A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations </a>, J. Int. Seq. 14 (2011) # 11.3.8

%H R. Cori, G. Hetyei, <a href="http://arxiv.org/abs/1306.4628">Counting genus one partitions and permutations</a>, arXiv:1306.4628 [math.CO], 2013.

%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.

%H A. Laradji, and A. Umar, <a href="http://dx.doi.org/10.1007/s00233-005-0553-6">Combinatorial results for semigroups of order-preserving full transformations</a>, Semigroup Forum 72 (2006), 51-62.

%F 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)^(2*k+2); coefficients of the numerators are the rows of the reverse triangle C(n, k)*C(n+1, k).

%F T(n,k) = C(n, k)*Sum_{j=0..(n-k)} C(n-j, k). - _Paul Barry_, Jan 12 2006

%F 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)]

%F 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). _Wolfdieter Lang_, Nov 13 2007.

%F 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_, Jan 24 2008

%F 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

%F T(n,k) = A003056(n+1,k+1)*C(n,k)^2/(k+1). - _Peter Luschny_, Oct 29 2011

%F T(n, k) = A007318(n, k)*A135278(n, k), n >= k >= 0. - _Wolfdieter Lang_, Jul 31 2017

%e The triangle T(n, k) begins:

%e n\k 0 1 2 3 4 5 6 7 8 9 ...

%e 0: 1

%e 1: 2 1

%e 2: 3 6 1

%e 3: 4 18 12 1

%e 4: 5 40 60 20 1

%e 5: 6 75 200 150 30 1

%e 6: 7 126 525 700 315 42 1

%e 7: 8 196 1176 2450 1960 588 56 1

%e 8: 9 288 2352 7056 8820 4704 1008 72 1

%e 9: 10 405 4320 17640 31752 26460 10080 1620 90 1

%e ... reformatted. - _Wolfdieter Lang_, Jul 31 2017

%e O.g.f. of 4th diagonal [4, 40,200, ...] is G(3, x) = 4*(1 + 3*x + x^2)/(1 - x)^7, from the n = 3 row [1, 3, 1] of A001263. See a comment above. - _Wolfdieter Lang_, Jul 31 2017

%p A103371 := (n,k) -> binomial(n,k)^2*(n+1)/(k+1);

%p seq(print(seq(A103371(n, k), k=0..n)), n=0..7); # _Peter Luschny_, Oct 19 2011

%t Flatten[Table[Binomial[n,n-k]Binomial[n+1,n-k],{n,0,10},{k,0,n}]] (* _Harvey P. Dale_, May 26 2014 *)

%o (Maxima) create_list(binomial(n,k)*binomial(n+1,k+1),n,0,12,k,0,n); /* _Emanuele Munarini_, Mar 11 2011 */

%o a103371 n k = a103371_tabl !! n !! k

%o a103371_row n = a103371_tabl !! n

%o a103371_tabl = map reverse a132813_tabl

%o -- _Reinhard Zumkeller_, Apr 04 2014

%o (MAGMA) /* As triangle */ [[Binomial(n,n-k)*Binomial(n+1,n-k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Aug 01 2017

%o (PARI) for(n=0,10, for(k=0,n, print1(binomial(n,k)*binomial(n+1,k+1), ", "))) \\ _G. C. Greubel_, Nov 09 2018

%Y Cf. A008459, A122899, A194595, A197653.

%Y Cf. A007318, A000894 (central terms), A132813 (mirrored).

%Y Cf. A000027, A002411, A004302, A108647, A134287, A135278, A001263.

%K easy,nonn,tabl

%O 0,2

%A _Paul Barry_, Feb 03 2005

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Last modified January 23 19:45 EST 2020. Contains 331175 sequences. (Running on oeis4.)