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Expansion of 1/((1-x)^2(1-x^2)(1-x^3)(1-x^5)) in powers of x.
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%I #25 Sep 06 2022 10:56:55

%S 1,2,4,7,11,17,25,35,48,64,84,108,137,171,211,258,312,374,445,525,616,

%T 718,832,959,1100,1256,1428,1617,1824,2050,2297,2565,2856,3171,3511,

%U 3878,4273,4697,5152,5639,6160,6716,7309,7940,8611,9324,10080,10881,11729

%N Expansion of 1/((1-x)^2(1-x^2)(1-x^3)(1-x^5)) in powers of x.

%C Partitions of n into parts 1, 2, 3, and 5. - _Joerg Arndt_, Jun 05 2014

%D Susan Elle, Ore extensions of global dimension 5, Abstract 1110-17-204, Abstracts Amer. Math. Soc., 36 (No. 2, 2015), p. 822.

%H <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,-1,1,0,-1,1,1,0,-2,1).

%F a(n) = round((n+1)*(n^3+23*n^2+173*n+451)/720). - _Tani Akinari_, Jun 05 2014

%F a(n) - 2*a(n-1) + a(n+3) + a(n+4) - 2*a(n+6) + a(n+7) = 1 if n == 3 (mod 5) else 0. - _Michael Somos_, Jun 05 2014

%F a(n) = a(-12 - n) for all n in Z. - _Michael Somos_, May 14 2015

%F a(n) - a(n-1) = A008669(n), a(n) - a(n-3) = A001304(n) for all n in Z. - _Michael Somos_, May 14 2015

%F Euler transform of length 5 sequence [ 2, 1, 1, 0, 1]. - _Michael Somos_, May 14 2015

%e G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 11*x^4 + 17*x^5 + 25*x^6 + 35*x^7 + ...

%t a[ n_] := Quotient[n (n + 12) (n^2 + 12 n + 52), 720] + 1; (* _Michael Somos_, Jun 05 2014 *)

%t a[ n_] := With[{m = If[ n < 0, -12 - n, n]}, SeriesCoefficient[ 1 / ((1 - x)^2*(1 - x^2)*(1 - x^3)*(1 - x^5)), {x, 0, m}]]; (* _Michael Somos_, Jun 05 2014 *)

%t Table[Round[(n + 1)*(n^3 + 23*n^2 + 173*n + 451)/720], {n, 0, 40}] (* _Wesley Ivan Hurt_, Jun 05 2014 *)

%t LinearRecurrence[{2,0,-1,-1,1,0,-1,1,1,0,-2,1},{1,2,4,7,11,17,25,35,48,64,84,108},50] (* _Harvey P. Dale_, Sep 06 2022 *)

%o (PARI) Vec(1/((1-x)^2*(1-x^2)*(1-x^3)*(1-x^5))+O(x^99)) \\ _Charles R Greathouse IV_, Sep 27 2012

%o (PARI) {a(n) = n * (n+12) * (n^2 + 12*n + 52) \ 720 + 1}; /* _Michael Somos_, Jun 05 2014 */

%o (PARI) {a(n) = if( n<0, n = -12 - n); polcoeff( 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^3) * (1 - x^5)) + x * O(x^n), n)}; /* _Michael Somos_, Jun 05 2014 */

%Y Cf. A001304, A008669.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Dec 11 1999