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A103924
Number of partitions of n into parts but with two kinds of parts of sizes 1,2,3,4 and 5.
9
1, 2, 5, 10, 20, 36, 64, 107, 177, 282, 443, 678, 1026, 1522, 2234, 3231, 4628, 6550, 9193, 12774, 17619, 24098, 32740, 44161, 59213, 78894, 104553, 137787, 180702, 235806, 306354, 396226, 510392, 654787, 836911, 1065734, 1352475, 1710535, 2156536, 2710318
OFFSET
0,2
COMMENTS
See A103923 for other combinatorial interpretations of a(n).
Convolution of A001401 and A000041. - Vaclav Kotesovec, Aug 28 2015
Also the sum of binomial (D(p), 5) over partitions p of n+15, 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 (reprinted 1962), p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
LINKS
FORMULA
G.f.: (product(1/(1-x^k), k=1..5)^2)*product(1/(1-x^j), j=6..infty).
a(n) = sum(A000710(n-5*j), j=0..floor(n/5)), n>=0.
a(n) ~ 3*n^(3/2) * exp(Pi*sqrt(2*n/3)) / (20*sqrt(2)*Pi^5). - Vaclav Kotesovec, Aug 28 2015
MAPLE
with(numtheory): a:= proc(n) a(n):=`if`(n=0, 1, add(add(d*`if`(d<6, 2, 1), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..40); # Alois P. Heinz, Sep 14 2014
MATHEMATICA
a[n_] := a[n] = If[n==0, 1, Sum[Sum[d*If[d<6, 2, 1], {d, Divisors[j]}] * a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 28 2015, after Alois P. Heinz *)
nmax=60; CoefficientList[Series[Product[1/(1-x^k), {k, 1, 5}] * Product[1/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 28 2015 *)
Table[Length@IntegerPartitions[n, All, Range@n~Join~Range@5], {n, 0, 39}] (* Robert Price, Jul 29 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+15, 5];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 30 2021 *)
CROSSREFS
Sixth column (m=5) of Fine-Riordan triangle A008951 and of triangle A103923, i.e. the p_2(n, m) array of the Gupta et al. reference.
Cf. A000712 (all parts of two kinds).
Sequence in context: A294536 A325650 A325720 * A160647 A103925 A160525
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Mar 24 2005
STATUS
approved