A350495 -id:A350495 - cp's OEIS frontend

A350249 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

1, 1, 1, 1, 1, 3, 7, 19, 43, 95, 189, 429, 1003, 2457, 6319, 16165, 41601, 107969, 280253, 737065, 1950865, 5201941, 13954313, 37593679, 101695957, 276296549, 753191093, 2061201397, 5658850121, 15583938539, 43040609115, 119182143639, 330841253283, 920550527585

Offset: 0

Ilya Gutkovskiy, Jan 28 2022

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A007576, A158092, A350495, A350880.

b:= proc(n) option remember; `if`(n=0, 1,
      expand((x^(n^2)+1+1/x^(n^2))*b(n-1)))
    end:
a:= n-> coeff(b(n),x,0):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 28 2022

Table[Coefficient[Product[x^(k^2) + 1 + 1/x^(k^2), {k, 1, n}], x, 0], {n, 0, 30}] (* Vaclav Kotesovec, Feb 05 2022 *)

Conjecture: a(n) ~ sqrt(5) * 3^(n + 1/2) / (2*sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, Feb 04 2022

A369389 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2))^2.

Original entry on oeis.org

1, 3, 9, 35, 141, 745, 3955, 23985, 155527, 1060941, 7393765, 53041015, 387815175, 2882682967, 21715452927, 165583974835, 1275674593889, 9918184576835, 77738274996385, 613753581566079, 4877383708962749, 38989308129231703, 313354624116918229, 2530796548734844153

Offset: 0

Ilya Gutkovskiy, Jan 22 2024

nonn

Cf. A047653, A156181, A350249, A350495.

Table[Coefficient[Product[(x^(k^2) + 1 + 1/x^(k^2))^2, {k, 1, n}], x, 0], {n, 0, 23}]

cp's OEIS Frontend

A350249 a(n) is the constant term in expansion of Product_{k=1..n} (x^(k^2) + 1 + 1/x^(k^2)).

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