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%I #17 Sep 30 2022 23:09:41
%S 0,1,5,16,42,100,228,512,1144,2544,5616,12288,26656,57408,122944,
%T 262144,556928,1179392,2490112,5242880,11010560,23069696,48235520,
%U 100663296,209713152,436203520,905965568,1879048192,3892322304,8053080064,16643014656,34359738368
%N a(n) = [x^n] ((1 - x)*x)/((1 - 2*x)^2*(2*x^2 - 2*x + 1)).
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,16,-8).
%F a(n) = Sum_{k=0..n} A323100(n - k, k).
%F a(n) = n! [x^n] exp(x)*(exp(x)*(2*x + 1) - sin(x) - cos(x))/2.
%F a(n) = 2*((2*n+2)*a(n-3) - (3*n+2)*a(n-2) + (2*n+1)*a(n-1))/n for n >= 4.
%F a(2^n - 1) = 2^(2^n + n - 2) if n>1. - _Michael Somos_, Sep 30 2022
%e G.f. = x + 5*x^2 + 16*x^3 + 42*x^4 + 100*x^5 + 228*x^6 + ... - _Michael Somos_, Sep 30 2022
%p ogf := ((1 - x)*x)/((1 - 2*x)^2*(2*x^2 - 2*x + 1));
%p ser := series(ogf, x, 32): seq(coeff(ser, x, n), n=0..31);
%t LinearRecurrence[{6,-14,16,-8}, {0,1,5,16}, 32] (* _Georg Fischer_, May 08 2021 *)
%o (PARI) {a(n) = if(n<0, 0, polcoeff( x*(1 - x) / ((1 - 2*x)^2*(1 - 2*x + 2*x^2)), n))}; /* _Michael Somos_, Sep 30 2022 */
%Y Antidiagonal sums of A323100.
%K nonn,easy
%O 0,3
%A _Peter Luschny_, Jan 12 2019
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