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A generalized partition function.
(Formerly M4077 N1693)
2

%I M4077 N1693 #30 Oct 17 2023 06:11:55

%S 1,6,9,13,19,37,58,97,143,227,328,492,688,992,1364,1903,2551,3473,

%T 4586,6097,7911,10333,13226,16988,21454,27172,33938,42437,52423,64833,

%U 79354,97130,117824,142930,172018,206925,247179,295105,350154,415124,489414,576540

%N A generalized partition function.

%D Hansraj Gupta, A generalization of the partition function. Proc. Nat. Inst. Sci. India 17, (1951), 231-238.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Vincenzo Librandi, <a href="/A002598/b002598.txt">Table of n, a(n) for n = 1..1000</a>

%H Hansraj Gupta, <a href="/A002597/a002597.pdf">A generalization of the partition function</a>, Proc. Nat. Inst. Sci. India 17, (1951). 231-238. [Annotated scanned copy]

%H <a href="/index/Rec#order_30">Index entries for linear recurrences with constant coefficients</a>, signature (1, 2, 1, 0, -9, -5, 2, 13, 21, -4, -17, -30, -13, 25, 28, 25, -13, -30, -17, -4, 21, 13, 2, -5, -9, 0, 1, 2, 1, -1).

%F G.f.: x*(3*x^32 -9*x^30 -10*x^29 -2*x^28 +29*x^27 +43*x^26 +9*x^25 -54*x^24 -107*x^23 -49*x^22 +76*x^21 +162*x^20 +125*x^19 -53*x^18 -189*x^17 -172*x^16 -11*x^15 +157*x^14 +166*x^13 +50*x^12 -81*x^11 -119*x^10 -49*x^9 +30*x^8 +55*x^7 +29*x^6 -8*x^5 -18*x^4 -9*x^3 +x^2 +5*x +1)/((x -1)^10*(x +1)^6*(x^2 +1)^4*(x^2 +x +1)^3). [_Colin Barker_, Oct 02 2012]

%t CoefficientList[Series[(3 x^32 - 9 x^30 - 10 x^29 - 2 x^28 + 29 x^27 + 43 x^26 + 9 x^25 - 54 x^24 - 107 x^23 - 49 x^22 + 76 x^21 + 162 x^20 + 125 x^19 - 53 x^18 - 189 x^17 - 172 x^16 - 11 x^15 + 157 x^14 + 166 x^13 + 50 x^12 - 81 x^11 - 119 x^10 - 49 x^9 + 30 x^8 + 55 x^7 + 29 x^6 - 8 x^5 - 18 x^4 - 9 x^3 + x^2 + 5 x + 1)/((x - 1)^10 (x + 1)^6 (x^2 + 1)^4 (x^2 + x + 1)^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Oct 13 2013 *)

%Y Essentially the same as A064349.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_

%E More terms from _Vincenzo Librandi_, Oct 13 2013