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A275327 Triangle read by rows, Riordan array (1, (2+(x-1)/(2*x^2)*(1-sqrt(1-4*x^2)))/ sqrt(1-4*x^2)).

%I A275327 #12 Mar 28 2020 05:29:22
%S A275327 1,0,1,0,1,1,0,3,2,1,0,2,7,3,1,0,10,10,12,4,1,0,5,33,25,18,5,1,0,35,
%T A275327 42,78,48,25,6,1,0,14,144,144,155,80,33,7,1,0,126,168,420,356,275,122,
%U A275327 42,8,1,0,42,610,723,1018,736,450,175,52,9,1
%N A275327 Triangle read by rows, Riordan array (1, (2+(x-1)/(2*x^2)*(1-sqrt(1-4*x^2)))/ sqrt(1-4*x^2)).
%e A275327 Table starts:
%e A275327 [n] [k=0,1,2,...] row sum
%e A275327 [0] [1] 1
%e A275327 [1] [0, 1] 1
%e A275327 [2] [0, 1, 1] 2
%e A275327 [3] [0, 3, 2, 1] 6
%e A275327 [4] [0, 2, 7, 3, 1] 13
%e A275327 [5] [0, 10, 10, 12, 4, 1] 37
%e A275327 [6] [0, 5, 33, 25, 18, 5, 1] 87
%e A275327 [7] [0, 35, 42, 78, 48, 25, 6, 1] 235
%e A275327 [8] [0, 14, 144, 144, 155, 80, 33, 7, 1] 578
%e A275327 [9] [0, 126, 168, 420, 356, 275, 122, 42, 8, 1] 1518
%p A275327 S := proc(n, k) option remember; local ecn:
%p A275327 if n = 0 then return n^k fi;
%p A275327 ecn := n -> n!/(iquo(n,2)!^2)/(iquo(n,2)+1);
%p A275327 add(ecn(i)*S(n-1,k-i), i=1..k-n+1) end:
%p A275327 A275327 := (n, k) -> S(k, n):
%p A275327 seq(seq(A275327(n, k),k=0..n),n=0..8);
%t A275327 (* The function RiordanArray is defined in A256893. *)
%t A275327 RiordanArray[1&, (2+(#-1)/(2#^2) (1-Sqrt[1-4#^2]))/Sqrt[1-4#^2]&, 11] // Flatten (* _Jean-François Alcover_, Jul 16 2019 *)
%o A275327 (Sage) # uses[riordan_array from A256893]
%o A275327 s = (2+(x-1)/(2*x^2)*(1-sqrt(1-4*x^2)))/sqrt(1-4*x^2)
%o A275327 riordan_array(1, s, 12)
%Y A275327 Cf. A057977 (column 1), A128899, A275328.
%K A275327 nonn,tabl
%O A275327 0,8
%A A275327 _Peter Luschny_, Aug 16 2016