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A373309
Number of binary partitions of n into three kinds of parts.
3
1, 3, 9, 19, 42, 78, 146, 246, 420, 668, 1068, 1620, 2470, 3618, 5310, 7546, 10746, 14910, 20706, 28134, 38262, 51090, 68238, 89706, 117964, 153012, 198468, 254332, 325914, 413214, 523778, 657606, 825444, 1027292, 1278060, 1577748, 1947062, 2386002, 2922702, 3557162, 4327644
OFFSET
0,2
COMMENTS
Submitted at the request of Joerg Arndt.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 521 terms from Paul D. Hanna)
FORMULA
G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following formulas.
(1) A(x) = 1 / Product_{n>=0} (1 - x^(2^n))^3, from Joerg Arndt, Fri Jun 21 2024.
(2) A(x) = Product_{n>=0} (1 + x^(2^n))^(3*(n+1)), deduced from a formula by Joerg Arndt in A018819.
(3) A(x) = A(x^2) / (1-x)^3.
(4) Convolution cube of A018819, which is the number of partitions of n into powers of 2.
EXAMPLE
G.f.: A(x) = 1 + 3*x + 9*x^2 + 19*x^3 + 42*x^4 + 78*x^5 + 146*x^6 + 246*x^7 + 420*x^8 + 668*x^9 + 1068*x^10 + 1620*x^11 + 2470*x^12 + ...
where A(x) = 1/((1-x)^3*(1-x^2)^3*(1-x^4)^3* ... * (1 - x^(2^k))^3 * ...).
MATHEMATICA
nmax = 40; CoefficientList[Series[1/Product[(1 - x^(2^k))^3, {k, 0, Log[2, nmax] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 25 2024 *)
PROG
(PARI) {a(n) = my(A = 1/prod(k=0, #binary(n), (1 - x^(2^k) +x*O(x^n))^3 )); polcoeff(A, n)}
for(n=0, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 24 2024
STATUS
approved