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A090181 Triangle of Narayana (A001263) with 0 <= k <= n, read by rows. 20
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 10, 20, 10, 1, 0, 1, 15, 50, 50, 15, 1, 0, 1, 21, 105, 175, 105, 21, 1, 0, 1, 28, 196, 490, 490, 196, 28, 1, 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1, 0, 1, 45, 540, 2520, 5292, 5292, 2520, 540, 45, 1, 0, 1, 55, 825, 4950, 13860 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Row sums: A000108 (Catalan). Columns give A000217, A002415, A006542, A006857. Row n=0: 1; row n=1: 0, 1; row n=2: 0, 1, 1; row n=3: 0, 1, 3, 1; row n=4: 0, 1, 6, 6, 1; row n=5: 0, 1, 10, 20, 10, 1; row n=6: 0, 1, 15, 50, 50, 15, 1.

Coefficient array of the polynomials P(n,x) = x^n*2F1(-n,-n+1;2;1/x). - Paul Barry, Nov 10 2008

Mirror image of triangle A131198. - Philippe Deléham, Dec 10 2008

Number of Dyck n-paths with exactly k peaks. - Peter Luschny, May 10 2014

LINKS

Indranil Ghosh, Rows 0..100, flattened

P. Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6

P. Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.

P. Barry, On a Generalization of the Narayana Triangle, J. Int. Seq. 14 (2011) # 11.4.5

Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.

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

FindStat - Combinatorial Statistic Finder, The number of peaks of a Dyck path, The number of double rises of a Dyck path, The number of valleys of a Dyck path, The number of left oriented leafs except the first one of a binary tree, The number of left tunnels of a Dyck path

Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011

FORMULA

Triangle T(n, k), read by rows, given by : [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is the operator defined in A084938 . T(0, 0) = 1, T(n, 0) = 0 for n>0, T(n, k) = C(n-1, k-1)*C(n, k-1)/k for k>0.

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000108(n), A006318(n), A047891(n+1), A082298(n), A082301(n), A082302(n), A082305(n), A082366(n), A082367(n) for x=0,1,2,3,4,5,6,7,8,9. - Philippe Deléham, Aug 10 2006

Sum_{k=0..n} x^(n-k)*T(n,k) = A090192(n+1), A000012(n), A000108(n), A001003(n), A007564(n), A059231(n), A078009(n), A078018(n), A081178(n), A082147(n), A082181(n), A082148(n), A082173(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. - Philippe Deléham, Oct 21 2006

Sum_[j, j>=0} T(n,j)*binomial(j,k) = A060693(n,k). - Philippe Deléham, May 04 2007

Sum_{k=0..n} T(n,k)*x^k*(x-1)^(n-k) = A000012(n), A006318(n), A103210(n), A103211(n), A133305(n), A133306(n), A133307(n), A133308(n), A133309(n) for x = 1, 2, 3, 4, 5, 6, 7, 8, 9, respectively. - Philippe Deléham, Oct 20 2007

Sum_{k=0..n}T(n,k)*10^k = A143749(n+1). - Philippe Deléham, Oct 14 2008

T(n,k) = Sum{j=0..n} (-1)^(j-k)*C(2n-j,j)*C(j,k)*A000108(n-j). - Paul Barry, Nov 10 2008

Sum_{k=0..n}T(n,k)*5^k*3^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008

Sum{k=0..n} T(n,k)*(-2)^k = A152681(n); Sum_{k=0..n} T(n,k)*(-1)^k = A105523(n). - Philippe Deléham, Feb 03 2009

Sum_{k=0..n} T(n,k)*2^(n+k) = A156017(n). - Philippe Deléham, Nov 27 2011

T(n, k) = C(n,n-k)*C(n-1,n-k)/(n-k+1). - Peter Luschny, May 10 2014

E.g.f.: 1+Integral((sqrt(t)*exp((1+t)*x)*BesselI(1,2*sqrt(t)*x))/x dx). - Peter Luschny, Oct 30 2014

EXAMPLE

From Paul Barry, Nov 10 2008: (Start)

Triangle begins

1;

0,  1;

0,  1,  1;

0,  1,  3,  1;

0,  1,  6,  6,  1;

0,  1, 10, 20, 10,  1;

0,  1, 15, 50, 50, 15,  1; (End)

MAPLE

A090181 := (n, k) -> binomial(n, n-k)*binomial(n-1, n-k)/(n-k+1): seq(print( seq(A090181(n, k), k=0..n)), n=0..5); # Peter Luschny, May 10 2014

# Alternatively:

egf := 1+int((sqrt(t)*exp((1+t)*x)*BesselI(1, 2*sqrt(t)*x))/x, x);

s := n -> n!*coeff(series(egf, x, n+2), x, n); seq(print(seq(coeff(s(n), t, j), j=0..n)), n=0..9); # Peter Luschny, Oct 30 2014

MATHEMATICA

Flatten[Table[Sum[(-1)^(j-k) * Binomial[2n-j, j] * Binomial[j, k] * CatalanNumber[n-j], {j, 0, n}], {n, 0, 11}, {k, 0, n}]] (* Indranil Ghosh, Mar 05 2017 *)

PROG

(Sage)

def A090181_row(n):

    U = [0]*(n+1)

    for d in DyckWords(n):

        U[d.number_of_peaks()] +=1

    return U

for n in range(8): A090181_row(n) # Peter Luschny, May 10 2014

(PARI)

c(n) = binomial(2*n, n)/ (n+1);

tabl(nn) = {for(n=0, nn, for(k=0, n, print1(sum(j=0, n, (-1)^(j-k) * binomial(2*n-j, j) * binomial(j, k) * c(n-j)), ", "); ); print(); ); };

tabl(11); \\ Indranil Ghosh, Mar 05 2017

(MAGMA) [[(&+[(-1)^(j-k)*Binomial(2*n-j, j)*Binomial(j, k)*Binomial(2*n-2*j, n-j)/(n-j+1): j in [0..n]]): k in [0..n]]: n in [0..10]];

CROSSREFS

Cf. A001263, A084938, A243752.

Sequence in context: A059045 A122935 A131198 * A256551 A144417 A085791

Adjacent sequences:  A090178 A090179 A090180 * A090182 A090183 A090184

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jan 19 2004

STATUS

approved

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Last modified May 20 16:03 EDT 2018. Contains 304339 sequences. (Running on oeis4.)