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A035382 Number of partitions of n into parts congruent to 1 mod 3. 4
1, 1, 1, 1, 2, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 11, 13, 15, 17, 19, 23, 26, 29, 33, 38, 42, 48, 54, 61, 68, 77, 85, 96, 107, 119, 132, 148, 163, 181, 201, 223, 245, 272, 299, 330, 363, 400, 438, 483, 529, 580, 635, 697, 760, 832, 908, 992, 1081, 1180, 1283, 1399, 1521 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(n) = A116373(3*n). - Reinhard Zumkeller, Feb 15 2006

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Joerg Arndt, Matters Computational (The Fxtbook), p. 350.

FORMULA

a(n) = 1/n*Sum_{k=1..n} A078181(k)*a(n-k), a(0) = 1.

G.f.: 1/prod(j>=0, 1-x^(1+3*j) ). - Emeric Deutsch, Mar 30 2006

From Joerg Arndt, Oct 02 2012: (Start)

G.f.: sum(n>=0, q^n/prod(k=1..n, 1-q^(3*k)) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(R*n)/prod(k=1..n, 1-q^(M*k) ) ) for partitions into parts R mod M (where R!=0).

G.f. sum(n>=0, q^(3*n^2-2*n) / prod(k=0..n-1, (1-q^(3*k+3))*(1-q^(3*k+1))) ); this is the special case of R=1, M=3 of the g.f. sum(n>=0, q^(M*n^2+(R-M)*n) / prod(k=0..n-1, (1-q^(M*k+M))*(1-q^(M*k+R))) ) for partitions into parts R mod M (where R!=0). (See Fxtbook link)

(End)

EXAMPLE

a(3) = 1 because we have [1,1,1];

a(4) = 2 because we have [1,1,1,1] and [4];

a(9) = 4 because we have [7,1,1], [4,4,1], [4,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].

1 + x + x^2 + x^3 + 2*x^4 + 2*x^5 + 2*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...

MAPLE

g:= 1/product(1-x^(1+3*j), j=0..50): gser:= series(g, x=0, 64): seq(coeff(gser, x, n), n=0..61); - Emeric Deutsch, Mar 30 2006

# second Maple program

b:= proc(n, i) option remember; `if`(n=0, 1,

      `if`(i<1, 0, b(n, i-3) +`if`(i>n, 0, b(n-i, i))))

    end:

a:= n-> b(n, 3*iquo(n, 3)+1):

seq (a(n), n=0..100);  # Alois P. Heinz, Oct 03 2012

CROSSREFS

Cf. A035386, A035451.

Sequence in context: A029075 A029052 A131795 * A094988 A173911 A076269

Adjacent sequences:  A035379 A035380 A035381 * A035383 A035384 A035385

KEYWORD

nonn

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified December 20 07:03 EST 2014. Contains 252241 sequences.