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%I #21 Feb 17 2021 03:54:57
%S 1,1,1,1,2,1,1,3,5,1,1,4,11,14,1,1,5,19,45,42,1,1,6,29,100,197,132,1,
%T 1,7,41,185,562,903,429,1,1,8,55,306,1257,3304,4279,1430,1,1,9,71,469,
%U 2426,8925,20071,20793,4862,1,1,10,89,680,4237,20076,65445,124996,103049,16796,1
%N Triangle derived from an array of f(x), Narayana polynomials.
%C Row sums = (1, 2, 4, 10, 31, 113, 466, 2129, 10641, 138628, 335379, 2702364,...)
%C Another version of triangle in A008550. - _Philippe Deléham_, Jan 13 2012
%C Another version of A243631. - _Philippe Deléham_, Sep 26 2014
%H G. C. Greubel, <a href="/A204057/b204057.txt">Rows n = 1..100 of the triangle, flattened</a>
%F The triangle is the set of antidiagonals of an array in which columns are f(x) of the Narayana polynomials; with column 1 = (1, 1, 1,...) column 2 = (1, 2, 3,..), column 3 = A028387, column 4 = A090197, then A090198, A090199,...
%F The array by rows is generated from production matrices of the form:
%F 1, (N-1)
%F 1, 1, (N-1)
%F 1, 1, 1, (N-1)
%F 1, 1, 1, 1, (N-1)
%F ...(infinite square matrices with the rest zeros); such that if the matrix is M, n-th term in row N is the upper left term of M^n.
%F From _G. C. Greubel_, Feb 16 2021: (Start)
%F T(n, k) = Hypergeometric2F1([1-k, -k], [2], n-k).
%F Sum_{k=1..n} T(n, k) = A132745(n) - 1. (End)
%e First few rows of the array =
%e 1,....1,....1,.....1,.....1,...; = A000012
%e 1.....2,....5,....14,....42,...; = A000108
%e 1,....3,...11,....45,...197,...; = A001003
%e 1,....4,...19,...100,...562,...; = A007564
%e 1,....5,...29,...185,..1257,...; = A059231
%e 1,....6,...41,...306,..2426,...; = A078009
%e ...
%e First few rows of the triangle =
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 3, 5, 1;
%e 1, 4, 11, 14, 1;
%e 1, 5, 19, 45, 42, 1;
%e 1, 6, 29, 100, 197, 132, 1;
%e 1, 7, 41, 185, 562, 903, 429, 1;
%e 1, 8, 55, 306, 1257, 3304, 4279, 1430, 1;
%e 1, 9, 71, 469, 2426, 8952, 20071, 20793, 4862, 1;
%e ...
%e Examples: column 4 of the array = A090197: (1, 14, 45, 100,...) = N(4,n) where N(4,x) is the 4th Narayana polynomial.
%e Term (5,3) = 29 is the upper left term of M^3, where M = the infinite square production matrix:
%e 1, 4, 0, 0, 0,...
%e 1, 1, 4, 0, 0,...
%e 1, 1, 1, 4, 0,...
%e 1, 1, 1, 1, 4,...
%e ... generating row 5, A059231: (1, 5, 29, 185,...).
%t Table[Hypergeometric2F1[1-k, -k, 2, n-k], {n,12}, {k,n}]//Flatten (* _G. C. Greubel_, Feb 16 2021 *)
%o (Sage)
%o def A204057(n, k): return 1 if n==0 else sum( binomial(n, j)^2*k^j*(n-j)/(n*(j+1)) for j in [0..n-1])
%o flatten([[A204057(k, n-k) for k in [1..n]] for n in [1..12]]) # _G. C. Greubel_, Feb 16 2021
%o (Magma)
%o A204057:= func< n, k | n eq 0 select 1 else (&+[ Binomial(n, j)^2*k^j*(n-j)/(n*(j+1)): j in [0..n-1]]) >;
%o [A204057(k, n-k): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Feb 16 2021
%Y Cf. A000108, A001003, A007564, A028387, A059231, A078009, A090197, A090198, A090199, A090200.
%Y Cf. A008550, A132745, A243631.
%K nonn,tabl
%O 1,5
%A _Gary W. Adamson_, Jan 09 2012
%E Corrected by _Philippe Deléham_, Jan 13 2012
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