A002598 A generalized partition function.

1, 6, 9, 13, 19, 37, 58, 97, 143, 227, 328, 492, 688, 992, 1364, 1903, 2551, 3473, 4586, 6097, 7911, 10333, 13226, 16988, 21454, 27172, 33938, 42437, 52423, 64833, 79354, 97130, 117824, 142930, 172018, 206925, 247179, 295105, 350154, 415124, 489414, 576540

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

References

• Hansraj Gupta, A generalization of the partition function. Proc. Nat. Inst. Sci. India 17, (1951), 231-238.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Hansraj Gupta, A generalization of the partition function, Proc. Nat. Inst. Sci. India 17, (1951). 231-238. [Annotated scanned copy]
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Essentially the same as A064349.

Programs

Mathematica

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 *)

Formula

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]

Extensions

More terms from Vincenzo Librandi, Oct 13 2013