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A003982 Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...

%I A003982 #39 Jan 29 2022 02:09:24
%S A003982 1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,
%T A003982 0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
%U A003982 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A003982 Table read by rows: 1 if x = y, 0 otherwise, where (x,y) = (0,0),(0,1),(1,0),(0,2),(1,1),(2,0),...
%C A003982 Also called the delta function.
%C A003982 From _Clark Kimberling_, Feb 07 2011: (Start)
%C A003982 In rectangular format, the infinite identity matrix and the weight array of A003783(n,k)=min{n,k}; in the accumulation chain
%C A003982 ... < A003982 < A003783 < A115262 < A185957 < ... . See A144112 for definitions of weight array and accumulation array. (End)
%H A003982 Antti Karttunen, <a href="/A003982/b003982.txt">Table of n, a(n) for n = 0..65537</a>
%H A003982 <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F A003982 n-th 1 is followed by 4*n-1 0's. In the sequence with flattened indices, the 1's are at positions listed in A046092.
%F A003982 G.f.: 1/(1 - x*y). E.g.f.: exp(x*y).
%F A003982 Considered as a linear sequence, expansion of q^(-1/2)*eta(q^8)^2/eta(q^4) in powers of q. If A(x) is the g.f., then B(a) = (q*A(a^2))^2 satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = u^2*w - v^3 - 4*v*w^2. Also, given g.f. A(x), then B(q) = q*A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1*u2^2*u6 - u1*u6^3 - u3^3*u2. - _Michael Somos_, Apr 13 2005
%F A003982 a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 if p>2. - _Michael Somos_, Jun 06 2005
%F A003982 a(n) = floor(sqrt(2*n+1)) - floor(sqrt(2*n)). - _Ridouane Oudra_, Oct 09 2020
%e A003982 Table begins
%e A003982   1;
%e A003982   0, 0;
%e A003982   0, 1, 0;
%e A003982   0, 0, 0, 0;
%e A003982   0, 0, 1, 0, 0;
%e A003982   ....
%e A003982 Northwest corner when formatted as a rectangular array:
%e A003982   1 0 0 0 0 0 0 0
%e A003982   0 1 0 0 0 0 0 0
%e A003982   0 0 1 0 0 0 0 0
%e A003982   0 0 0 1 0 0 0 0
%e A003982   0 0 0 0 1 0 0 0
%t A003982 f[n_,k_]:=0; f[n_,n_]:=1;
%t A003982 TableForm[Table[f[n,k],{n,1,10},{k,1,10}]] (* array *)
%t A003982 Table[f[n-k+1,k],{n,10},{k,n,1,-1}]//Flatten (*sequence *)
%t A003982 Table[Join[{1},Table[0,4n-1]],{n,10}]//Flatten (* _Harvey P. Dale_, Dec 21 2016 *)
%o A003982 (PARI) {a(n) = issquare(2*n + 1)}; /* _Michael Somos_, Apr 13 2005 */
%o A003982 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^8 + A)^2 / eta(x^4 + A), n))};
%o A003982 (PARI) A(i,j)=i==j
%Y A003982 Characteristic function of A001844. Antidiagonal sums and main diagonal is A000012.
%Y A003982 Cf. also A286100.
%K A003982 tabl,nonn,nice,easy
%O A003982 0,1
%A A003982 _Marc LeBrun_