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A035382 Number of partitions of n into parts congruent to 1 mod 3. 16
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)

a(n) ~ GAMMA(1/3) * exp(sqrt(2*n)*Pi/3) / (2*sqrt(3) * (2*Pi*n)^(2/3)). - Vaclav Kotesovec, Feb 26 2015

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

MATHEMATICA

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-3] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[n, 3*Quotient[n, 3]+1]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 23 2015, after Alois P. Heinz *)

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 3 14:15 EST 2016. Contains 278745 sequences.