A373309 Number of binary partitions of n into three kinds of parts.

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

Author

Paul D. Hanna, Jun 24 2024

Keywords

• nonn

Comments

Submitted at the request of Joerg Arndt.

Examples

			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 * ...).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 521 terms from Paul D. Hanna)

Crossrefs

Cf. A018819, A171238, A373308.

Programs

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 *)

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),", "))

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.