A240944 - OEIS

%I #17 Aug 29 2021 02:08:01

%S 1,1,1,1,3,5,7,9,15,28,45,66,99,164,269,422,651,1028,1654,2637,4149,

%T 6522,10350,16467,26091,41205,65174,103339,163833,259361,410376,

%U 649827,1029543,1630725,2581848,4087797,6473832,10253370,16237375,25711316,40714953,64478427,102111230,161701086,256062990,405499697,642156651

%N Number of compositions of n into square parts k^2 where there are k sorts of part k^2.

%H Alois P. Heinz, <a href="/A240944/b240944.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1/(1 - sum(k>=1, k * x^(k^2)) ).

%p a:= proc(n) option remember; `if`(n=0, 1, `if`(n<0, 0,

%p       add(k*a(n-k^2), k=1..isqrt(n))))

%p     end:

%p seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 04 2014

%t a[n_] := a[n] = If[n == 0, 1, If[n<0, 0, Sum[k*a[n-k^2], {k, Sqrt[n]}]]];

%t Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Aug 29 2021, after _Alois P. Heinz_ *)

%o (PARI) N=66; x='x+O('x^N); Vec(1/(1 - sum(k=1, 1+sqrtint(N), k * x^(k^2))) )

%Y Cf. A006456 (compositions into squares).

%K nonn

%O 0,5

%A _Joerg Arndt_, Aug 04 2014