A369433 -id:A369433 - cp's OEIS frontend

A369734 Number of solutions to 1^2k_1 + 2^2k_2 + ... + n^2*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

0, 1, 1, 1, 1, 3, 8, 17, 35, 79, 177, 409, 995, 2475, 6336, 16078, 41401, 107304, 279550, 736032, 1950000, 5199850, 13950852, 37576658, 101670863, 276228026, 753114256, 2060995699, 5658663542, 15583477334, 43039552072, 119179072495, 330836308272, 920537185436

Offset: 0

Ilya Gutkovskiy, Jan 30 2024

nonn

Cf. A007576, A063866, A350249, A369433, A369628, A369735.

Table[Coefficient[Product[(x^(k^2) + 1 + 1/x^(k^2)), {k, 1, n}], x, 1], {n, 0, 32}]

a(n) = [x^1] Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

A368478 a(n) = [x^n] Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 22, 88, 236, 568, 1147, 1999, 3030, 4333, 4793, 5620, 4628, 9353, 40048, 312436, 1638550, 7894061, 30866741, 105342660, 323671449, 863529149, 2152428536, 4750103818, 9796453345, 18572505806, 32181800057, 55712628236

Offset: 0

Ilya Gutkovskiy, Jan 25 2024

nonn

Cf. A000583, A316706, A369358, A369433, A369437.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1, b(n, i-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-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 25 2024

a(34)-a(38) from Alois P. Heinz, Jan 25 2024

A369495 a(n) = [x^n] Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 11, 25, 56, 129, 313, 748, 1831, 4584, 11581, 29555, 76398, 199191, 522746, 1382158, 3676072, 9828631, 26412174, 71310248, 193346941, 526302232, 1437895915, 3941671019, 10839118898, 29893768219, 82671240995, 229213877639, 637049969282

Offset: 0

Ilya Gutkovskiy, Jan 24 2024

nonn

Cf. A000217, A316706, A351002, A369344, A369433.

b:= proc(n, i) option remember; (m-> `if`(n>m, 0,
     `if`(n=m, 1, b(abs(n-i*(i+1)/2), i-1)+b(n, i-1)+
        b(n+i*(i+1)/2, i-1))))((2+(3+i)*i)*i/6)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..32);  # Alois P. Heinz, Jan 24 2024

Table[Coefficient[Product[x^(k (k + 1)/2) + 1 + 1/x^(k (k + 1)/2), {k, 1, n}], x, n], {n, 0, 32}]

cp's OEIS Frontend

A369734 Number of solutions to 1^2k_1 + 2^2k_2 + ... + n^2*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

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Formula

A368478 a(n) = [x^n] Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

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Extensions

A369495 a(n) = [x^n] Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

cp's OEIS Frontend

A369734 Number of solutions to 1^2*k_1 + 2^2*k_2 + ... + n^2*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A368478 a(n) = [x^n] Product_{k=1..n} (x^(k^4) + 1 + 1/x^(k^4)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Extensions

A369495 a(n) = [x^n] Product_{k=1..n} (x^(k*(k+1)/2) + 1 + 1/x^(k*(k+1)/2)).

Views

Author

Keywords

Crossrefs

Programs

Maple

Mathematica

A369734 Number of solutions to 1^2k_1 + 2^2k_2 + ... + n^2*k_n = 1, where k_i are from {-1,0,1}, i=1..n.

A369495 a(n) = [x^n] Product_{k=1..n} (x^(k(k+1)/2) + 1 + 1/x^(k(k+1)/2)).