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 A090181 Triangle of Narayana (A001263) with 0 <= k <= n, read by rows. 40
 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 Number of Dyck n-paths with exactly k peaks. - Peter Luschny, May 10 2014 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 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. 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 EXAMPLE 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; 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 *) 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 *) 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 (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 (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 Mirror image of triangle A131198. A000108 (row sums, Catalan). Columns: A000217, A002415, A006542, A006857. Cf. A001263, A084938, A243752. 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_{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 Sequence in context: A348210 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 December 5 13:14 EST 2023. Contains 367591 sequences. (Running on oeis4.)