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A001400 Number of partitions of n into at most 4 parts.
(Formerly M0627 N0229)
32
1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 206, 225, 249, 270, 297, 321, 351, 378, 411, 441, 478, 511, 551, 588, 632, 672, 720, 764, 816, 864, 920, 972, 1033, 1089, 1154, 1215, 1285, 1350, 1425, 1495 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Molien series for 4-dimensional representation of S_4 [Nebe, Rains, Sloane, Chap. 7].

Also number of pure 2-complexes on 4 nodes with n multiple 2-simplexes - Vladeta Jovovic, Dec 27 1999

Also number of different integer triangles with perimeter <= n+3. Also number of different scalene integer triangles with perimeter <= n+9. - Reinhard Zumkeller, May 12 2002

a(n) = coefficient of q^n in the expansion of (m choose 4)_q as m goes to infinity. - Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 21 2002

For n>3: also number of partitions of n into parts <= 4: a(n)=A026820(n,4). [From Reinhard Zumkeller, Jan 21 2010]

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, row m=4 of Q(m,n) table; p. 120, P(n,4).

H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 2.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 275.

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).

D. E. Knuth, The Art of Computer Programming, vol. 4, Fascicle 3, Generating All Combinations and Partitions, Addison-Wesley, 2005, Section 7.2.1.4., p. 56, exercise 31.

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

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

F. Ellermann, Illustration for A001400, A061924

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 353

Gerzson Keri and Patric R. J. Ostergard, The Number of Inequivalent (2R+3,7)R Optimal Covering Codes, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.7.

Jon Perry, More Partition Functions

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index to sequences with linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

FORMULA

G.f.: 1/((1-x)*(1-x^2)*(1-x^3)*(1-x^4)).

a(n)=1+(a(n-2)+a(n-3)+a(n-4))-(a(n-5)+a(n-6)+a(n-7))+a(n-9). - Norman J. Meluch (norm(AT)iss.gm.com), Mar 09 2000

P(n, 4) = 1/288( 2*n^3 + 6*n^2 - 9*n - 13 + (9*n+9)*pcr{1, -1}(2, n)-32*pcr{1, -1, 0}(3, n)-36*pcr{1, 0, -1, 0}(4, n)) (see Comtet).

Let c(n)=sum(i=0, floor(n/3), 1+ceil((n-3*i-1)/2)), then a(n) = sum(i=0, floor(n/4), 1+ceil((n-4*i-1)/2)+c(n-4*i-3)). - Jon Perry, Jun 27 2003

Euler transform of finite sequence [1, 1, 1, 1].

(n choose 4)_q=(q^n-1)*(q^(n-1)-1)*(q^(n-2)-1)*(q^(n-3)-1)/((q^4-1)*(q^3-1)*(q^2-1)*(q-1)).

a(n) = round(((n+4)^3 + 3(n+4)^2 -9(n+4)((n+4) mod 2))/144). - Washington Bomfim, Jul 03 2012

EXAMPLE

(4 choose 4)_q = q^4 + q^3 + q^2 + q + 1, (5 choose 4)_q = q^4 + q^3 + q^2 + q + 1, (6 choose 4)_q = q^8 + q^7 + 2*q^6 + 2*q^5 + 3*q^4 + 2*q^3 + 2*q^2 + q + 1, (7 choose 4) = q^12 + q^11 + 2*q^10 + 3*q^9 + 4*q^8 + 4*q^7 + 5*q^6 + 4*q^5 + 4*q^4 + 3*q^3 + 2*q^2 + q + 1 so the coefficient of q^0 converges to 1, q^1 to 1, q^2 to 2 and so on.

MAPLE

A001400 := n->if n mod 2 = 0 then round(n^2*(n+3)/144); else round((n-1)^2*(n+5)/144); fi;

with(combstruct):ZL5:=[S, {S=Set(Cycle(Z, card<5))}, unlabeled]:seq(count(ZL5, size=n), n=0..55); - Zerinvary Lajos, Sep 24 2007

A001400:=-(-z**8+z**9+2*z**4-z**7-1-z)/(z**2+1)/(z**2+z+1)/(z+1)**2/(z-1)**4; [Conjectured by Simon Plouffe in his 1992 dissertation. Gives sequence except for an initial 1.]

B:=[S, {S = Set(Sequence(Z, 1 <= card), card <=4)}, unlabelled]: seq(combstruct[count](B, size=n), n=0..55); # Zerinvary Lajos, Mar 21 2009

MATHEMATICA

CoefficientList[ Series[ 1/((1 - x)*(1 - x^2)*(1 - x^3)*(1 - x^4)), {x, 0, 65} ], x ]

LinearRecurrence[{1, 1, 0, 0, -2, 0, 0, 1, 1, -1}, {1, 1, 2, 3, 5, 6, 9, 11, 15, 18}, 80] (* Vladimir Joseph Stephan Orlovsky, Feb 17 2012 *)

a[n_] := Sum[Floor[(n - j - 3*k + 2)/2], {j, 0, Floor[n/4]}, {k, j, Floor[(n - j)/3]}]; Table[a[n], {n, 0, 55}] (* L. Edson Jeffery, Jul 31 2014 *)

PROG

(MAGMA) K:=Rationals(); M:=MatrixAlgebra(K, 4); q1:=DiagonalMatrix(M, [1, -1, 1, -1]); p1:=DiagonalMatrix(M, [1, 1, -1, -1]); q2:=DiagonalMatrix(M, [1, 1, 1, -1]); h:=M![1, 1, 1, 1, 1, 1, -1, -1, 1, -1, 1, -1, 1, -1, -1, 1]/2; G:=MatrixGroup<4, K|q1, q2, h>; MolienSeries(G);

(PARI) a(n) = round(((n+4)^3 + 3*(n+4)^2 -9*(n+4)*((n+4)% 2))/144) \\ Washington Bomfim, Jul 03 2012

(Haskell)

a001400 n = a001400_list !! n

a001400_list = scanl1 (+) a005044_list -- Reinhard Zumkeller, Feb 28 2013

CROSSREFS

Essentially same as A026810. Partial sums of A005044. Cf. A070083.

a(n)=A008284(n+4, 4), n >= 0.

Cf. A008619, A001399, A001401, A117486.

First differences of A002621.

Sequence in context: A028309 A242717 A026810 * A008773 A008772 A008771

Adjacent sequences:  A001397 A001398 A001399 * A001401 A001402 A001403

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers, Aug 29 2000

STATUS

approved

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Last modified September 2 15:16 EDT 2014. Contains 246361 sequences.