A002622 Number of partitions of at most n into at most 5 parts. (Formerly M1053 N0395)

1, 2, 4, 7, 12, 19, 29, 42, 60, 83, 113, 150, 197, 254, 324, 408, 509, 628, 769, 933, 1125, 1346, 1601, 1892, 2225, 2602, 3029, 3509, 4049, 4652, 5326, 6074, 6905, 7823, 8837, 9952, 11178, 12520, 13989, 15591, 17338, 19236, 21298, 23531, 25949, 28560, 31378, 34412

Offset

0,2

References

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

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

E. Fix and J. L. Hodges, Jr., Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312.

E. Fix and J. L. Hodges, Significance probabilities of the Wilcoxon test, Annals Math. Stat., 26 (1955), 301-312. [Annotated scanned copy]

Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 0, 2, -1).

Formula

G.f.: 1/[(1+x^2)*(1-x^3)*(1-x)^4*(1-x^5)*(1+x)^2]. (Corrected Mar 31 2018)

a(n)= 2*a(n-1) -a(n-3) -a(n-5) +2*a(n-8) -a(n-11) -a(n-13) +2*a(n-15) -a(n-16).

G.f.: 1 / ((1 - x)^2 * (1 - x^2) * (1 - x^3) * (1 - x^4) * (1 - x^5)). - Michael Somos, Apr 24 2014

Euler transform of length 5 sequence [ 2, 1, 1, 1, 1]. - Michael Somos, Apr 24 2014

a(n) = a(n-1) + A001401(n). - Michael Somos, Apr 24 2014

a(n) = round((n+1)*(6*n^4+234*n^3+3326*n^2+20674*n+50651+675*(-1)^n)/86400). - Tani Akinari, May 05 2014

Example

G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 29*x^6 + 42*x^7 + 60*x^8 + ...
a(2) = 4 with partitions 0, 1, 2, 1+1. a(3) = 7 with partitions 0, 1, 2, 1+1, 3, 2+1, 1+1+1. - Michael Somos, Apr 24 2014

Mathematica

CoefficientList[Series[1/((1 - x)^2 (1 - x^2) (1 - x^3) (1 - x^4) (1 - x^5)), {x, 0, 100}], x] (* Vincenzo Librandi, Apr 25 2014 *)
LinearRecurrence[{2, 0, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 0, 2, -1},  {1, 2, 4, 7, 12, 19, 29, 42, 60, 83, 113, 150, 197, 254, 324, 408},  48] (* Georg Fischer, Feb 27 2019 *)

Prog

(PARI) x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 5, 1-x^i))) \\ Altug Alkan, Mar 30 2018

Crossrefs

Cf. A001401 (first differences). Column 5 of A092905.

Sequence in context: A333311 A266464 A103231 * A363276 A035301 A035297

Adjacent sequences: A002619 A002620 A002621 * A002623 A002624 A002625

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved