A080247 Formal inverse of triangle A080246. Unsigned version of A080245.
1, 2, 1, 6, 4, 1, 22, 16, 6, 1, 90, 68, 30, 8, 1, 394, 304, 146, 48, 10, 1, 1806, 1412, 714, 264, 70, 12, 1, 8558, 6752, 3534, 1408, 430, 96, 14, 1, 41586, 33028, 17718, 7432, 2490, 652, 126, 16, 1, 206098
Offset: 0
Examples
Triangle starts: [0] 1 [1] 2, 1 [2] 6, 4, 1 [3] 22, 16, 6, 1 [4] 90, 68, 30, 8, 1 [5] 394, 304, 146, 48, 10, 1 [6] 1806, 1412, 714, 264, 70, 12, 1 ... From _Gary W. Adamson_, Jul 25 2011: (Start) n-th row = top row of M^n, M = the following infinite square production matrix: 2, 1, 0, 0, 0, ... 2, 2, 1, 0, 0, ... 2, 2, 2, 1, 0, ... 2, 2, 2, 2, 1, ... ... (End)
Links
- Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
- Paul Barry, Laurent Biorthogonal Polynomials and Riordan Arrays, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
- Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
- Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 21, 29.
- James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
- P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
- Shishuo Fu and Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
- W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 5.
Programs
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Maple
A080247:=(n,k)->(k+1)*add(binomial(n+1,k+j+1)*binomial(n+j,j),j=0..n-k)/(n+1): seq(seq(A080247(n,k),k=0..n),n=0..9);
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Mathematica
Clear[w] w[n_, k_] /; k < 0 || k > n := 0 w[0,0]=1 ; w[n_, k_] /; 0 <= k <= n && !n == k == 0 := w[n, k] = w[n-1,k-1] + w[n-1,k] + w[n,k+1] Table[w[n,k],{n,0,10},{k,0,n}] (* David Callan, Jul 03 2006 *) T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -1]; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Peter Luschny, Jan 08 2018 *) -
Maxima
T(n,k):=((k+1)*sum(2^m*binomial(n+1,m)*binomial(n-k-1,n-k-m),m,0,n-k))/(n+1); /* Vladimir Kruchinin, Jan 10 2022 */
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Sage
def A080247_row(n): @cached_function def prec(n, k): if k==n: return 1 if k==0: return 0 return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1)) return [(-1)^(n-k)*prec(n, k) for k in (1..n)] for n in (1..10): print(A080247_row(n)) # Peter Luschny, Mar 16 2016
Formula
G.f.: 2/(2+y*x-y+y*(x^2-6*x+1)^(1/2))/y/x. - Vladeta Jovovic, Feb 16 2003
Essentially same triangle as triangle T(n,k), n > 0 and k > 0, read by rows; given by [0, 2, 1, 2, 1, 2, 1, 2, 1, 2, ...] DELTA A000007 where DELTA is Deléham's operator defined in A084938.
T(n, k) = T(n-1, k-1) + 2*Sum_{j>=0} T(n-1, k+j) with T(0, 0) = 1 and T(n, k)=0 if k < 0. - Philippe Deléham, Jan 19 2004
T(n, k) = (k+1)*Sum_{j=0..n-k} (binomial(n+1, k+j+1)*binomial(n+j, j))/(n+1). - Emeric Deutsch, May 31 2004
Recurrence: T(0,0)=1; T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n,k+1). - David Callan, Jul 03 2006
T(n, k) = binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -1). - Peter Luschny, Jan 08 2018
T(n,k) = (k+1)/(n+1)*Sum_{m=0..n-k} 2^m*binomial(n+1,m)*binomial(n-k-1,n-k-m). - Vladimir Kruchinin, Jan 10 2022
From Peter Bala, Sep 16 2024: (Start)
Riordan array (S(x), x*S(x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the g.f. of the large Schröder numbers A006318.
Comments