OFFSET
0,2
COMMENTS
Also number of partitions of 2*n+4 with exactly 4 odd parts. - Vladeta Jovovic, Jan 12 2005
Also the sum of binomial (D(p), 4) over partitions p of n+10, where D(p) is the number of different part sizes in p. - Emily Anible, Jun 09 2018
REFERENCES
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
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
Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms
FORMULA
Euler transform of 2 2 2 2 1 1 1...
G.f.: 1/((1-x)(1-x^2)(1-x^3)(1-x^4)*Product_{k>=1} (1-x^k)).
a(n) = Sum_{j=0..floor(n/4)} A000098(n-4*j), n >= 0.
a(n) ~ sqrt(3)*n * exp(Pi*sqrt(2*n/3)) / (8*Pi^4). - Vaclav Kotesovec, Aug 18 2015
EXAMPLE
a(2) = 5 because we have 2, 2', 1+1, 1+1', 1'+1'.
MAPLE
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr(n-> `if`(n<5, 2, 1)): seq(a(n), n=0..40); # Alois P. Heinz, Sep 08 2008
MATHEMATICA
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; a = etr[If[#<5, 2, 1]&]; Table[a[n], {n, 0, 39}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[1/((1-x)(1-x^2)(1-x^3)(1-x^4))*Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 18 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@4], {n, 0, 39}] (* Robert Price, Jul 28 2020 *)
T[n_, 0] := PartitionsP[n];
T[n_, m_] /; (n >= m (m + 1)/2) := T[n, m] = T[n - m, m - 1] + T[n - m, m];
T[_, _] = 0;
a[n_] := T[n + 10, 4];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by Emeric Deutsch, Mar 22 2005
STATUS
approved