A373309 - OEIS

%I #10 Jun 25 2024 19:12:27

%S 1,3,9,19,42,78,146,246,420,668,1068,1620,2470,3618,5310,7546,10746,

%T 14910,20706,28134,38262,51090,68238,89706,117964,153012,198468,

%U 254332,325914,413214,523778,657606,825444,1027292,1278060,1577748,1947062,2386002,2922702,3557162,4327644

%N Number of binary partitions of n into three kinds of parts.

%C Submitted at the request of _Joerg Arndt_.

%H Alois P. Heinz, <a href="/A373309/b373309.txt">Table of n, a(n) for n = 0..10000</a> (first 521 terms from Paul D. Hanna)

%F G.f. A(x) = Sum_{n>=0} a(n) * x^n satisfies the following formulas.

%F (1) A(x) = 1 / Product_{n>=0} (1 - x^(2^n))^3, from _Joerg Arndt_, Fri Jun 21 2024.

%F (2) A(x) = Product_{n>=0} (1 + x^(2^n))^(3*(n+1)), deduced from a formula by _Joerg Arndt_ in A018819.

%F (3) A(x) = A(x^2) / (1-x)^3.

%F (4) Convolution cube of A018819, which is the number of partitions of n into powers of 2.

%e 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 + ...

%e where A(x) = 1/((1-x)^3*(1-x^2)^3*(1-x^4)^3* ... * (1 - x^(2^k))^3 * ...).

%t 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 *)

%o (PARI) {a(n) = my(A = 1/prod(k=0,#binary(n), (1 - x^(2^k) +x*O(x^n))^3 )); polcoeff(A,n)}

%o for(n=0,40,print1(a(n),", "))

%Y Cf. A018819, A171238, A373308.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 24 2024