A090181 Triangle of Narayana (A001263) with 0 <= k <= n, read by rows.
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
Offset: 0
Examples
Triangle starts: [0] 1; [1] 0, 1; [2] 0, 1, 1; [3] 0, 1, 3, 1; [4] 0, 1, 6, 6, 1; [5] 0, 1, 10, 20, 10, 1; [6] 0, 1, 15, 50, 50, 15, 1; [7] 0, 1, 21, 105, 175, 105, 21, 1; [8] 0, 1, 28, 196, 490, 490, 196, 28, 1; [9] 0, 1, 36, 336, 1176, 1764, 1176, 336, 36, 1;
Links
- Indranil Ghosh, Rows 0..100, flattened
- Yu Hin Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
- Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
- Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv preprint arXiv:1107.5490 [math.CO], 2011.
- Paul 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.
- Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
- Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
- Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 25, 29.
- Paul Barry and Aoife Hennessy, A Note on Narayana Triangles and Related Polynomials, Riordan Arrays, and MIMO Capacity Calculations, J. Int. Seq. 14 (2011), Article 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.
Crossrefs
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
Programs
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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]];
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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 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 T := proc(n, k) option remember; if k = n or k = 1 then 1 elif k < 1 then 0 else (2*n/k - 1) * T(n-1, k-1) + T(n-1, k) fi end: for n from 0 to 8 do seq(T(n, k), k = 0..n) od; # Peter Luschny, Dec 31 2024
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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 *) p[0, ] := 1; p[1, x] := x; p[n_, x_] := ((2 n - 1) (1 + x) p[n - 1, x] - (n - 2) (x - 1)^2 p[n - 2, x]) / (n + 1); Table[CoefficientList[p[n, x], x], {n, 0, 9}] // TableForm (* Peter Luschny, Apr 26 2022 *) -
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 -
Python
from functools import cache @cache def Trow(n): if n == 0: return [1] if n == 1: return [0, 1] if n == 2: return [0, 1, 1] A = Trow(n - 2) + [0, 0] B = Trow(n - 1) + [1] for k in range(n - 1, 1, -1): B[k] = (((B[k] + B[k - 1]) * (2 * n - 1) - (A[k] - 2 * A[k - 1] + A[k - 2]) * (n - 2)) // (n + 1)) return B for n in range(10): print(Trow(n)) # Peter Luschny, May 02 2022 -
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
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_{j>=0} T(n,j)*binomial(j,k) = A060693(n,k). - Philippe Deléham, May 04 2007
Sum_{k=0..n} T(n,k)*10^k = A143749(n+1). - Philippe Deléham, Oct 14 2008
From Paul Barry, Nov 10 2008: (Start)
Coefficient array of the polynomials P(n,x) = x^n*2F1(-n,-n+1;2;1/x).
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*C(2n-j,j)*C(j,k)*A000108(n-j). (End)
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
G.f.: (1+x-x*y-sqrt((1-x*(1+y))^2-4*y*x^2))/(2*x). - Alois P. Heinz, Nov 28 2021, edited by Ron L.J. van den Burg, Dec 19 2021
T(n, k) = [x^k] (((2*n - 1)*(1 + x)*p(n-1, x) - (n - 2)*(x - 1)^2*p(n-2, x))/(n + 1)) with p(0, x) = 1 and p(1, x) = x. - Peter Luschny, Apr 26 2022
Recursion based on rows (see the Python program):
T(n, k) = (((B(k) + B(k-1))*(2*n - 1) - (A(k) - 2*A(k-1) + A(k-2))*(n-2))/(n+1)), where A(k) = T(n-2, k) and B(k) = T(n-1, k), for n >= 3. # Peter Luschny, May 02 2022
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