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 A254594 Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x. 4
 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, 83, 123, 102, 147, 123, 174, 147, 204, 174, 237, 204, 274, 237, 314, 274, 358, 314, 406, 358, 458, 406, 514, 458, 575, 514, 640, 575, 710, 640, 785, 710, 865, 785, 950 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Partitions of n into parts of size 3 and size 4 and two kinds of parts of size 2. 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. Euler transform of length 4 sequence [ 0, 2, 1, 1]. LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Index entries for linear recurrences with constant coefficients, signature (0,2,1,0,-2,-2,0,1,2,0,-1). FORMULA G.f.: 1 / (1 - 2*x^2 - x^3 + 2*x^5 + 2*x^6 - x^8 - 2*x^9 + x^11). a(n) = -a(-11-n) for all n in Z. a(n+3) - a(n) = 0 if n even else floor((n+7)^2 / 16). 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. a(n) - a(n-2) = A005044(n+3) for all n in Z. a(n) + a(n-1) = A001400(n) for all n in Z. a(n) + a(n-2) = A165188(n+1) for all n in Z. a(n) = A115264(n) - A115264(n-1) for all n in Z. 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 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 EXAMPLE 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 + ... MATHEMATICA a[ n_] := Quotient[ n^3 + If[ OddQ[n], 12 n^2 + 33 n + 54, 21 n^2 + 132 n + 288], 288]; 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}]]; 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]; 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 *) PROG (PARI) {a(n) = (n^3 + if(n%2, 12*n^2 + 33*n + 54, 21*n^2 + 132*n + 288)) \ 288}; (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)}; (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 CROSSREFS Cf. A001400, A002620, A005044, A115264, A165188. Cf. A241526, A000601, A181120. Sequence in context: A276055 A252866 A008796 * A280948 A325345 A079966 Adjacent sequences: A254591 A254592 A254593 * A254595 A254596 A254597 KEYWORD nonn,easy AUTHOR Michael Somos, Feb 02 2015 STATUS approved

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Last modified August 14 10:40 EDT 2024. Contains 375159 sequences. (Running on oeis4.)