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%I #20 Jan 08 2023 10:13:45
%S 1,1,1,1,3,1,1,3,5,1,1,3,9,7,1,1,3,9,19,9,1,1,3,9,27,33,11,1,1,3,9,27,
%T 65,51,13,1,1,3,9,27,81,131,73,15,1,1,3,9,27,81,211,233,99,17,1,1,3,9,
%U 27,81,243,473,379,129,19,1,1,3,9,27,81,243,665,939,577,163,21,1
%N Riordan array (1/(1-x), x*(1+2*x)).
%C Inverse is A110292.
%H G. C. Greubel, <a href="/A110291/b110291.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = [x^n]( x^k*(1+2*x)^k/(1-x) ).
%F Sum_{k=0..n} T(n, k) = A000975(n+1)).
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A052947(n+1).
%F From _G. C. Greubel_, Jan 05 2023: (Start)
%F T(n, 0) = T(n, n) = 1.
%F T(n, n-1) = A005408(n-1).
%F T(2*n, n) = T(2*n+1), n) = A000244(n).
%F T(2*n, n+1) = A066810(n+1).
%F T(2*n, n-1) = A000244(n-1).
%F T(2*n+1, n+1) = A001047(n+1).
%F Sum_{k=0..n} (-1)^k * T(n, k) = A077912(n).
%F Sum_{k=0..n} 2^k * T(n, k) = A014335(n+2).
%F Sum_{k=0..n} 3^k * T(n, k) = A180146(n).
%F Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A077890(n). (End)
%e Rows begin
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 3, 5, 1;
%e 1, 3, 9, 7, 1;
%e 1, 3, 9, 19, 9, 1;
%e 1, 3, 9, 27, 33, 11, 1;
%e 1, 3, 9, 27, 65, 51, 13, 1;
%e 1, 3, 9, 27, 81, 131, 73, 15, 1;
%t F[k_]:= CoefficientList[Series[x^k*(1+2*x)^k/(1-x), {x,0,40}], x];
%t A110291[n_, k_]:= F[k][[n+1]];
%t Table[A110291[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 05 2023 *)
%o (Magma)
%o R<x>:=PowerSeriesRing(Rationals(), 30);
%o F:= func< k | Coefficients(R!( x^k*(1+2*x)^k/(1-x) )) >;
%o A110291:= func< n,k | F(k)[n-k+1] >;
%o [A110291(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jan 05 2023
%o (SageMath)
%o def p(k,x): return x^k*(1+2*x)^k/(1-x)
%o def A110291(n,k): return ( p(k,x) ).series(x, 30).list()[n]
%o flatten([[A110291(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 05 2023
%Y Cf. A000244, A001045, A001047, A005408, A014335.
%Y Cf. A066810, A077890, A077912, A110292, A180146.
%Y Cf. A000975 (row sums), A052947 (diagonal sums).
%K easy,nonn,tabl
%O 0,5
%A _Paul Barry_, Jul 18 2005
%E a(30) and following corrected by _Georg Fischer_, Oct 11 2022
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