A254594 Expansion of 1 / ((1 - x^2)^2 * (1 - x^3) * (1 - x^4)) in powers of x.

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

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