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

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