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%I #14 Mar 18 2021 23:59:10
%S 1,1,0,3,3,0,0,0,15,15,0,0,0,0,5,100,100,5,0,0,0,0,0,105,840,840,105,
%T 0,0,0,0,0,0,42,1764,8589,8589,1764,42,0,0,0,0,0,0,7,1792,29232,
%U 104104,104104,29232,1792,7,0,0,0,0,0,0,0,1080,53505,508680,1463760,1463760,508680,53505,1080
%N G.f.: A(x,y) = (1-y)^2 * Sum_{n>=0} (2*n+1) * y^n * (1 + x*(1-y)^2 )^(n*(n+1)/2).
%C Compare to: Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)/2) = Product_{n>=1} (1 - x^n)^3.
%C G.f.: A(x,y) = Sum_{n>=0} Sum_{k=0..2*n+1} T(n,k) * x^n*y^k.
%H Paul D. Hanna, <a href="/A303650/b303650.txt">Table of n, a(n) for n = 0..2651 of rows 0..50 flattened</a>
%F Row sums = 2*(2*n+1)!/(2^n*n!) = 2*A001147(n+1) for n >= 0, where A001147 is the odd double factorials.
%e G.f.: A(x,y) = (1 + y) + (3*y + 3*y^2)*x + (15*y^2 + 15*y^3)*x^2 + (5*y^2 + 100*y^3 + 100*y^4 + 5*y^5)*x^3 + (105*y^3 + 840*y^4 + 840*y^5 + 105*y^6)*x^4 + (42*y^3 + 1764*y^4 + 8589*y^5 + 8589*y^6 + 1764*y^7 + 42*y^8)*x^5 + (7*y^3 + 1792*y^4 + 29232*y^5 + 104104*y^6 + 104104*y^7 + 29232*y^8 + 1792*y^9 + 7*y^10)*x^6 + (1080*y^4 + 53505*y^5 + 508680*y^6 + 1463760*y^7 + 1463760*y^8 + 508680*y^9 + 53505*y^10 + 1080*y^11)*x^7 + (405*y^4 + 63495*y^5 + 1433205*y^6 + 9504540*y^7 + 23457780*y^8 + 23457780*y^9 + 9504540*y^10 + 1433205*y^11 + 63495*y^12 + 405*y^13)*x^8 + (90*y^4 + 53255*y^5 + 2737090*y^6 + 37539550*y^7 + 192046210*y^8 + 422352880*y^9 + 422352880*y^10 + 192046210*y^11 + 37539550*y^12 + 2737090*y^13 + 53255*y^14 + 90*y^15)*x^9 + (9*y^4 + 32835*y^5 + 3860661*y^6 + 103586637*y^7 + 998951415*y^8 + 4199275509*y^9 + 8443603509*y^10 + 8443603509*y^11 + 4199275509*y^12 + 998951415*y^13 + 103586637*y^14 + 3860661*y^15 + 32835*y^16 + 9*y^17)*x^10 + ...
%e This table begins:
%e [1, 1];
%e [0, 3, 3, 0];
%e [0, 0, 15, 15, 0, 0];
%e [0, 0, 5, 100, 100, 5, 0, 0];
%e [0, 0, 0, 105, 840, 840, 105, 0, 0, 0];
%e [0, 0, 0, 42, 1764, 8589, 8589, 1764, 42, 0, 0, 0];
%e [0, 0, 0, 7, 1792, 29232, 104104, 104104, 29232, 1792, 7, 0, 0, 0];
%Y Cf. A303651, A303652.
%K nonn,tabf
%O 0,4
%A _Paul D. Hanna_, Apr 30 2018
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