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A103371 Number triangle T(n,k)=C(n,n-k)C(n+1,n-k). 21
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; text; internal format)
OFFSET

0,2

COMMENTS

Columns include A000027, A002411, A004302, A108647, A134287. 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 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.

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]

The matrix inverse starts

1;

-2,1;

9,-6,1;

-76,54,-12,1;

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

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

636447,-460026,110775,-13300,945,-42,1; - R. J. Mathar, Mar 29 2013

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

LINKS

Reinhard Zumkeller, Rows n = 0..125 of table, flattened

P. Barry, A. Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations , J. Int. Seq. 14 (2011) # 11.3.8

R. Cori, G. Hetyei, Counting genus one partitions and permutations, arXiv preprint arXiv:1306.4628 [math.CO], 2013.

Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.

A. Laradji, and A. Umar, Combinatorial results for semigroups of order-preserving full transformations, Semigroup Forum 72 (2006), 51-62.

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, 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). Wolfdieter 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, 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

T(n, k) = A007318(n, k)*A135278(n, k), n >= k >= 0. - Wolfdieter Lang, Jul 31 2017

EXAMPLE

The triangle T(n, k) begins:

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

0:   1

1:   2   1

2:   3   6    1

3:   4  18   12     1

4:   5  40   60    20     1

5:   6  75  200   150    30     1

6:   7 126  525   700   315    42     1

7:   8 196 1176  2450  1960   588    56    1

8:   9 288 2352  7056  8820  4704  1008   72  1

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

...  reformatted. - Wolfdieter Lang, Jul 31 2017

O.g.f. of 4-th 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

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

MATHEMATICA

Flatten[Table[Binomial[n, n-k]Binomial[n+1, n-k], {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, May 26 2014 *)

PROG

(Maxima) create_list(binomial(n, k)*binomial(n+1, k+1), n, 0, 12, k, 0, n); /* Emanuele Munarini, Mar 11 2011 */

(Haskell)

a103371 n k = a103371_tabl !! n !! k

a103371_row n = a103371_tabl !! n

a103371_tabl = map reverse a132813_tabl

-- Reinhard Zumkeller, Apr 04 2014

(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

CROSSREFS

Cf. A008459, A122899, A194595, A197653.

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

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

Sequence in context: A115196 A093346 A115597 * A120257 A059298 A214306

Adjacent sequences:  A103368 A103369 A103370 * A103372 A103373 A103374

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, Feb 03 2005

STATUS

approved

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Last modified February 20 21:53 EST 2018. Contains 299387 sequences. (Running on oeis4.)