cp's OEIS Frontend

A111178 Number of partitions of n into positive numbers one less than a square.

%I A111178 #27 Oct 30 2021 18:00:43
%S A111178 1,0,0,1,0,0,1,0,1,1,0,1,1,0,1,2,1,1,2,1,1,2,1,2,4,1,2,4,1,2,5,2,4,5,
%T A111178 2,5,5,2,6,7,4,6,7,5,6,8,6,8,12,6,9,13,6,10,15,8,14,15,9,16,16,10,18,
%U A111178 21,14,19,22,16,20,24,19,25,30,20,27,33,21,29,39,26,37,40,28,42,42,31,48
%N A111178 Number of partitions of n into positive numbers one less than a square.
%C A111178 Also limiting form of the number of representations of n into k positive squares for k decreasing from n to 1, or Table[Count[SumOfSquaresRepresentations[k,n], {a_,__}/;a>0], {n,100,100}, {k,100,40,-1}]. (_Franklin T. Adams-Watters_: replacing k^2 ones by the value k^2 changes the count by k^2-1).
%C A111178 a(n) = A243148(2n,n). - _Alois P. Heinz_, May 30 2014
%H A111178 Alois P. Heinz, <a href="/A111178/b111178.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%F A111178 G.f.: Product_{k>=2} 1/(1-x^(k^2-1)).
%p A111178 b:= proc(n, i) option remember;
%p A111178       `if`(n=0, 1, `if`(i<2, 0, b(n, i-1)+
%p A111178       `if`(i^2>n+1, 0, b(n+1-i^2, i))))
%p A111178     end:
%p A111178 a:= n-> b(n, isqrt(n)):
%p A111178 seq(a(n), n=0..100);  # _Alois P. Heinz_, May 30 2014
%t A111178 nn = 100; CoefficientList[Series[Product[1/(1 - x^(k^2 - 1)), {k, 2, nn}], {x, 0, nn}], x] (* corrected by _T. D. Noe_, Feb 22 2012 *)
%t A111178 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<2, 0, b[n, i-1] + If[i^2>n+1, 0, b[n+1-i^2, i]]]]; a[n_] := b[n, Round[Sqrt[n]]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 16 2017, after _Alois P. Heinz_ *)
%o A111178 (Haskell)
%o A111178 a111178 = p $ tail a005563_list where
%o A111178    p _          0 = 1
%o A111178    p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
%o A111178 -- _Reinhard Zumkeller_, Apr 02 2014
%Y A111178 Cf. A001156, A078134.
%Y A111178 Cf. A005563, A319799.
%K A111178 easy,nonn
%O A111178 0,16
%A A111178 _Wouter Meeussen_, Oct 22 2005