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%I #25 Sep 08 2022 08:46:11
%S 1,0,2,1,4,2,7,4,11,7,16,11,23,16,31,23,41,31,53,41,67,53,83,67,102,
%T 83,123,102,147,123,174,147,204,174,237,204,274,237,314,274,358,314,
%U 406,358,458,406,514,458,575,514,640,575,710,640,785,710,865,785,950
%N Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.
%C Partitions of n into parts of size 3 and size 4 and two kinds of parts of size 2.
%C The number of quadruples of integers [x, u, v, w] which satisfy x > u > v > w >=0, n+5 = x+u, u+v >= x+w, and x+u+v+w is even.
%C Euler transform of length 4 sequence [ 0, 2, 1, 1].
%H G. C. Greubel, <a href="/A254594/b254594.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1,0,-2,-2,0,1,2,0,-1).
%F G.f.: 1 / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11).
%F a(n) = -a(-11-n) for all n in Z.
%F a(n+3) - a(n) = 0 if n even else floor((n+7)^2 / 16).
%F 0 = a(n) - 2*a(n+2) - a(n+3) + 2*a(n+5) + 2*a(n+6) - a(n+8) - 2*a(n+9) + a(n+11) for all n in Z.
%F a(n) - a(n-2) = A005044(n+3) for all n in Z.
%F a(n) + a(n-1) = A001400(n) for all n in Z.
%F a(n) + a(n-2) = A165188(n+1) for all n in Z.
%F a(n) = A115264(n) - A115264(n-1) for all n in Z.
%F a(2*n) - a(2*n-6) = a(2*n+3) - a(2*n-3) = A002620(n+2) for all n in Z. - _Michael Somos_, Feb 11 2015
%F a(n) = (2*n^3+33*n^2+181*n+234+3*(3*n^2+33*n+86)*(-1)^n+84*(-1)^((2*n+1-(-1)^n)/4)-96*((1+(-1)^n)*floor(((2*n+9+(-1)^n-6*(-1)^((2*n+3+(-1)^n)/4))/24))+(1-(-1)^n)*floor(((2*n+5+(-1)^n-6*(-1)^((2*n-1+(-1)^n)/4))/24))))/576. - _Luce ETIENNE_, May 22 2015
%e G.f. = 1 + 2*x^2 + x^3 + 4*x^4 + 2*x^5 + 7*x^6 + 4*x^7 + 11*x^8 + 7*x^9 + ...
%t a[ n_] := Quotient[ n^3 + If[ OddQ[n], 12 n^2 + 33 n + 54, 21 n^2 + 132 n + 288], 288];
%t a[ n_] := Module[{s = 1, m = n}, If[ n < 0, s = -1; m = -11 - n]; s SeriesCoefficient[ 1 / ((1 - x^2)^2 (1 - x^3) (1 - x^4)), {x, 0, m}]];
%t a[ n_] := Length @ FindInstance[ {x > u, u > v, v > w, w >= 0, x + u == n + 5, u + v >= x + w, x + u + v + w == 2 k}, {x, u, v, w, k}, Integers, 10^9];
%t CoefficientList[Series[1 / (1 - 2 x^2 - x^3 + 2 x^5 + 2 x^6 - x^8 - 2 x^9 + x^11), {x, 0, 60}], x] (* _Vincenzo Librandi_, Feb 03 2015 *)
%o (PARI) {a(n) = (n^3 + if(n%2, 12*n^2 + 33*n + 54, 21*n^2 + 132*n + 288)) \ 288};
%o (PARI) {a(n) = my(s=1); if( n<0, s=-1; n=-11-n); s * polcoeff( 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) + x * O(x^n), n)};
%o (Magma) I:=[1,0,2,1,4,2,7,4,11,7,16]; [n le 11 select I[n] else 2*Self(n-2)+Self(n-3)-2*Self(n-5)-2*Self(n-6)+Self(n-8)+2*Self(n-9)-Self(n-11): n in [1..60]]; // _Vincenzo Librandi_, Feb 03 2015
%Y Cf. A001400, A002620, A005044, A115264, A165188.
%Y Cf. A241526, A000601, A181120.
%K nonn,easy
%O 0,3
%A _Michael Somos_, Feb 02 2015
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