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A306462 Number of ways to write n as C(2w,2) + C(x+2,3) + C(y+3,4) + C(z+4,5), where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!), w is a positive integer and x,y,z are nonnegative integers.

%I A306462 #34 Feb 18 2019 14:28:51
%S A306462 1,3,3,1,1,4,7,6,2,2,6,8,5,1,2,9,11,5,1,4,9,12,7,2,4,10,12,7,4,6,10,
%T A306462 11,6,5,5,10,15,8,4,7,11,14,9,4,5,11,14,6,6,10,15,12,5,7,8,11,14,7,5,
%U A306462 6,11,14,12,11,6,11,15,12,7,9,18,21,12,5,5,15,19,11,3,5
%N A306462 Number of ways to write n as C(2w,2) + C(x+2,3) + C(y+3,4) + C(z+4,5), where C(n,k) denotes the binomial coefficient n!/(k!*(n-k)!), w is a positive integer and x,y,z are nonnegative integers.
%C A306462 Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 1, 4, 5, 14, 19.
%C A306462 We have verified a(n) > 0 for all n = 1..5*10^6.
%C A306462 See also A306471 and A306477 for similar conjectures.
%H A306462 Zhi-Wei Sun, <a href="/A306462/b306462.txt">Table of n, a(n) for n = 1..10000</a>
%e A306462 a(1) = 1 with 1 = C(2,2) + C(2,3) + C(3,4) + C(4,5).
%e A306462 a(4) = 1 with 4 = C(2,2) + C(3,3) + C(4,4) + C(5,5).
%e A306462 a(5) = 1 with 5 = C(2,2) + C(4,3) + C(3,4) + C(4,5).
%e A306462 a(14) = 1 with 14 = C(4,2) + C(3,3) + C(4,4) + C(6,5).
%e A306462 a(19) = 1 with 19 = C(6,2) + C(4,3) + C(3,4) + C(4,5).
%t A306462 f[m_,n_]:=f[m,n]=Binomial[m+n-1,m];
%t A306462 HQ[n_]:=HQ[n]=IntegerQ[Sqrt[8n+1]]&&Mod[Sqrt[8n+1],4]==3;
%t A306462 tab={};Do[r=0;Do[If[f[5,z]>=n,Goto[cc]];Do[If[f[4,y]>=n-f[5,z],Goto[bb]];Do[If[f[3,x]>=n-f[5,z]-f[4,y],Goto[aa]];If[HQ[n-f[5,z]-f[4,y]-f[3,x]],r=r+1],{x,0,n-1-f[5,z]-f[4,y]}];Label[aa],{y,0,n-1-f[5,z]}];Label[bb],{z,0,n-1}];Label[cc];tab=Append[tab,r],{n,1,80}];Print[tab]
%Y A306462 Cf. A000217, A000292, A000332, A000384, A000389, A000797, A262813, A306460, A306471, A306477.
%K A306462 nonn
%O A306462 1,2
%A A306462 _Zhi-Wei Sun_, Feb 17 2019