A369438
a(n) = [x^(n^3)] Product_{k=1..n} (x^(k^3) + 1 + 1/x^(k^3)).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 2, 3, 4, 11, 24, 46, 106, 238, 537, 1318, 3007, 7027, 18199, 43202, 105900, 279860, 688474, 1741235, 4641670, 11790546, 30529486, 82306963, 213852619, 563866091, 1531711961, 4047719392, 10835966180, 29624064007, 79423421277, 215083283638
Offset: 0
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b:= proc(n, i) option remember; `if`(n>(i*(i+1)/2)^2, 0,
`if`(i=0, 1, b(n, i-1)+b(n+i^3, i-1)+b(abs(n-i^3), i-1)))
end:
a:= n-> b(n^3, n):
seq(a(n), n=0..35); # Alois P. Heinz, Jan 23 2024
-
Table[Coefficient[Product[(x^(k^3) + 1 + 1/x^(k^3)), {k, 1, n}], x, n^3], {n, 0, 34}]
A367416
Triangle read by rows: T(n,k) = number of solutions to +- 1^k +- 2^k +- 3^k +- ... +- n^k is a k-th power, n >= 2.
Original entry on oeis.org
4, 8, 1, 16, 1, 32, 0, 2, 64, 6, 128, 8, 256, 16, 4, 512, 26, 1024, 17, 10, 2048, 67, 4, 3, 4096, 100, 10, 8192, 137, 34, 6, 16384, 426, 28, 1, 32768, 661, 96, 6, 65536, 1351, 146, 16, 8, 131072, 2637, 230, 15, 262144, 3831, 258, 40, 524288, 8095, 1130, 50
Offset: 2
Triangle begins:
k = 1 2 3 4 5
n= 2: 4;
n= 3: 8, 1;
n= 4: 16, 1;
n= 5: 32, 0, 2;
n= 6: 64, 6;
n= 7: 128, 8;
n= 8: 256, 16, 4;
n= 9: 512, 26;
n=10: 1024, 17, 10;
n=11: 2048, 67, 4, 3;
n=12: 4096, 100, 10;
n=13: 8192, 137, 34, 6;
n=14: 16384, 426, 28, 1;
n=15: 32768, 661, 96, 6;
n=16: 65536, 1351, 146, 16, 8;
n=17: 131072, 2637, 230, 15;
n=18: 262144, 3831, 258, 40;
n=19: 524288, 8095, 1130, 50;
n=20: 1048576, 15241, 854, 77, 6;
...
The T(6,2) = 6 solutions are:
- 1^2 - 2^2 + 3^2 - 4^2 + 5^2 + 6^2 = 49 = 7^2,
- 1^2 - 2^2 + 3^2 + 4^2 + 5^2 - 6^2 = 9 = 3^2,
- 1^2 - 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 81 = 9^2,
+ 1^2 - 2^2 + 3^2 - 4^2 - 5^2 + 6^2 = 1 = 1^2,
+ 1^2 + 2^2 - 3^2 + 4^2 + 5^2 - 6^2 = 1 = 1^2,
+ 1^2 + 2^2 + 3^2 - 4^2 - 5^2 + 6^2 = 9 = 3^2.
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f(k,u)=my(x=0,v=vector(#u));for(i=1,#u,u[i]=if(u[i]==0,-1,1);v[i]=i^k);u*v~
is(k,u)=my(x=f(k,u));ispower(x,k)
T(n,k)=my(u=vector(n,i,[0,1]),nbsol=0);if(k%2==1,u[1]=[1,1]);forvec(X=u,if(is(k,X),nbsol++));if(k%2==1,nbsol*=2);nbsol
A368348
a(n) = [x^(n^4)] Product_{k=1..n} (x^(k^4) + 1/x^(k^4)).
Original entry on oeis.org
1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 19, 26, 0, 0, 40, 129, 0, 0, 616, 785, 0, 0, 4080, 9309, 0, 0, 44775, 72659, 0, 0, 430297, 781505, 0, 0, 3934457, 7765047, 0, 0, 44740433, 78818429, 0, 0, 463089552, 900950811, 0, 0, 5344766190, 9806206864, 0, 0
Offset: 0
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b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
b(abs(n-i^4), i-1)+b(n+i^4, i-1))))(i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30)
end:
a:= n-> `if`(irem(n, 4)>1, 0, b(n^4, n)):
seq(a(n), n=0..43); # Alois P. Heinz, Jan 25 2024
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b[n_, i_] := b[n, i] = Function[m, If[n > m, 0, If[n == m, 1, b[Abs[n-i^4], i-1] + b[n+i^4, i-1]]]][i*(i+1)*(2*i+1)*(3*i^2+3*i-1)/30];
a[n_] := If[Mod[n, 4] > 1, 0, b[n^4, n]];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Feb 14 2025, after Alois P. Heinz *)
Showing 1-3 of 3 results.
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