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a(n) = [x^((n*(n+1)/2)^2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).
2

%I #19 Oct 29 2018 12:09:42

%S 1,1,2,7,28,262,3428,52289,1147221,30161625,893291633,30894822277,

%T 1214415301634,52617692115135,2528123847871538,133088227043557512,

%U 7574733515354756765,466116310963215784930,30810712157925101729430,2173319693639115252360852,163247410881483617710298406

%N a(n) = [x^((n*(n+1)/2)^2)] Product_{k=1..n} Sum_{m>=0} x^(k*m^2).

%C Also the number of nonnegative integer solutions (a_1, a_2, ... , a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = (n*(n+1)/2)^2.

%e 1* 0^2 + 2*0^2 + 3*0^2 + 4*5^2 = 100.

%e 1* 0^2 + 2*6^2 + 3*2^2 + 4*2^2 = 100.

%e 1* 1^2 + 2*2^2 + 3*3^2 + 4*4^2 = 100.

%e 1* 1^2 + 2*2^2 + 3*5^2 + 4*2^2 = 100.

%e 1* 1^2 + 2*4^2 + 3*1^2 + 4*4^2 = 100.

%e 1* 1^2 + 2*6^2 + 3*3^2 + 4*0^2 = 100.

%e 1* 2^2 + 2*4^2 + 3*0^2 + 4*4^2 = 100.

%e 1* 2^2 + 2*4^2 + 3*4^2 + 4*2^2 = 100.

%e 1* 3^2 + 2*0^2 + 3*3^2 + 4*4^2 = 100.

%e 1* 3^2 + 2*0^2 + 3*5^2 + 4*2^2 = 100.

%e 1* 3^2 + 2*6^2 + 3*1^2 + 4*2^2 = 100.

%e 1* 4^2 + 2*0^2 + 3*4^2 + 4*3^2 = 100.

%e 1* 4^2 + 2*2^2 + 3*2^2 + 4*4^2 = 100.

%e 1* 4^2 + 2*4^2 + 3*4^2 + 4*1^2 = 100.

%e 1* 4^2 + 2*6^2 + 3*2^2 + 4*0^2 = 100.

%e 1* 5^2 + 2*0^2 + 3*5^2 + 4*0^2 = 100.

%e 1* 5^2 + 2*2^2 + 3*1^2 + 4*4^2 = 100.

%e 1* 5^2 + 2*4^2 + 3*3^2 + 4*2^2 = 100.

%e 1* 5^2 + 2*6^2 + 3*1^2 + 4*0^2 = 100.

%e 1* 6^2 + 2*0^2 + 3*0^2 + 4*4^2 = 100.

%e 1* 6^2 + 2*0^2 + 3*4^2 + 4*2^2 = 100.

%e 1* 7^2 + 2*2^2 + 3*3^2 + 4*2^2 = 100.

%e 1* 7^2 + 2*4^2 + 3*1^2 + 4*2^2 = 100.

%e 1* 8^2 + 2*0^2 + 3*0^2 + 4*3^2 = 100.

%e 1* 8^2 + 2*2^2 + 3*2^2 + 4*2^2 = 100.

%e 1* 8^2 + 2*4^2 + 3*0^2 + 4*1^2 = 100.

%e 1* 9^2 + 2*0^2 + 3*1^2 + 4*2^2 = 100.

%e 1*10^2 + 2*0^2 + 3*0^2 + 4*0^2 = 100.

%e So a(4) = 28.

%Y Cf. A000122, A000537, A300446, A320932.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 29 2018

%E a(17)-a(20) from _Alois P. Heinz_, Oct 29 2018