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A080247 Formal inverse of triangle A080246. Unsigned version of A080245.

%I A080247 #79 Apr 18 2025 13:15:21
%S A080247 1,2,1,6,4,1,22,16,6,1,90,68,30,8,1,394,304,146,48,10,1,1806,1412,714,
%T A080247 264,70,12,1,8558,6752,3534,1408,430,96,14,1,41586,33028,17718,7432,
%U A080247 2490,652,126,16,1,206098
%N A080247 Formal inverse of triangle A080246. Unsigned version of A080245.
%C A080247 Row sums are little Schroeder numbers A001003. Diagonal sums are generalized Fibonacci numbers A006603. Columns include A006318, A006319, A006320, A006321.
%C A080247 T(n,k) is the number of dissections of a convex (n+3)-gon by nonintersecting diagonals with exactly k diagonals emanating from a fixed vertex. Example: T(2,1)=4 because the dissections of the convex pentagon ABCDE having exactly one diagonal emanating from the vertex A are: {AC}, {AD}, {AC,EC} and {AD,BD}. - _Emeric Deutsch_, May 31 2004
%C A080247 For more triangle sums, see A180662, see the Schroeder triangle A033877 which is the mirror of this triangle. - _Johannes W. Meijer_, Jul 15 2013
%H A080247 Michael De Vlieger, <a href="/A080247/b080247.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H A080247 Paul Barry, <a href="http://arxiv.org/abs/1311.2292">Laurent Biorthogonal Polynomials and Riordan Arrays</a>, arXiv preprint arXiv:1311.2292 [math.CA], 2013.
%H A080247 Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry3/barry422.html">Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles</a>, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
%H A080247 Paul Barry, <a href="https://arxiv.org/abs/2504.09719">Notes on Riordan arrays and lattice paths</a>, arXiv:2504.09719 [math.CO], 2025. See pp. 21, 29.
%H A080247 James East and Nicholas Ham, <a href="https://arxiv.org/abs/1811.05735">Lattice paths and submonoids of Z^2</a>, arXiv:1811.05735 [math.CO], 2018.
%H A080247 P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of noncrossing configurations</a>, Discrete Math. 204 (1999), 203-229.
%H A080247 Shishuo Fu and Yaling Wang, <a href="https://arxiv.org/abs/1908.03912">Bijective recurrences concerning two Schröder triangles</a>, arXiv:1908.03912 [math.CO], 2019.
%H A080247 W.-j. Woan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Woan/woan655.html">The Lagrange inversion formula and divisibility properties</a>, JIS 10 (2007) 07.7.8, example 5.
%F A080247 G.f.: 2/(2+y*x-y+y*(x^2-6*x+1)^(1/2))/y/x. - _Vladeta Jovovic_, Feb 16 2003
%F A080247 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.
%F A080247 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
%F A080247 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
%F A080247 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
%F A080247 T(n, k) = binomial(n, k)*hypergeom([k - n, n + 1], [k + 2], -1). - _Peter Luschny_, Jan 08 2018
%F A080247 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
%F A080247 From _Peter Bala_, Sep 16 2024: (Start)
%F A080247 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.
%F A080247 For integer m and n >= 1, (m + 2)*[x^n] S(x)^(m*n) = m*[x^n] (1/S(-x))^((m+2)*n). For cases of this identity see A103885 (m = 1), A333481 (m = 2) and A370102 (m = 3). (End)
%e A080247 Triangle starts:
%e A080247 [0]    1
%e A080247 [1]    2,    1
%e A080247 [2]    6,    4,   1
%e A080247 [3]   22,   16,   6,   1
%e A080247 [4]   90,   68,  30,   8,  1
%e A080247 [5]  394,  304, 146,  48, 10,  1
%e A080247 [6] 1806, 1412, 714, 264, 70, 12, 1
%e A080247 ...
%e A080247 From _Gary W. Adamson_, Jul 25 2011: (Start)
%e A080247 n-th row = top row of M^n, M = the following infinite square production matrix:
%e A080247   2, 1, 0, 0, 0, ...
%e A080247   2, 2, 1, 0, 0, ...
%e A080247   2, 2, 2, 1, 0, ...
%e A080247   2, 2, 2, 2, 1, ...
%e A080247   ... (End)
%p A080247 A080247:=(n,k)->(k+1)*add(binomial(n+1,k+j+1)*binomial(n+j,j),j=0..n-k)/(n+1):
%p A080247 seq(seq(A080247(n,k),k=0..n),n=0..9);
%t A080247 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 A080247 T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -1];
%t A080247 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Peter Luschny_, Jan 08 2018 *)
%o A080247 (Sage)
%o A080247 def A080247_row(n):
%o A080247     @cached_function
%o A080247     def prec(n, k):
%o A080247         if k==n: return 1
%o A080247         if k==0: return 0
%o A080247         return prec(n-1,k-1)-2*sum(prec(n,k+i-1) for i in (2..n-k+1))
%o A080247     return [(-1)^(n-k)*prec(n, k) for k in (1..n)]
%o A080247 for n in (1..10): print(A080247_row(n)) # _Peter Luschny_, Mar 16 2016
%o A080247 (Maxima)
%o A080247 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 */
%Y A080247 Cf. A000007, A033877 (mirror), A084938.
%K A080247 nonn,tabl
%O A080247 0,2
%A A080247 _Paul Barry_, Feb 15 2003