A375293 -id:A375293 - cp's OEIS frontend

A375292 Expansion of 1/sqrt((1 - x + x^3)^2 + 4*x^4).

1, 1, 1, 0, -3, -8, -14, -15, 1, 51, 146, 261, 286, -24, -1029, -2975, -5375, -5930, 591, 22014, 63886, 115947, 128183, -14595, -486466, -1413161, -2569868, -2840890, 361667, 10972167, 31861581, 57980426, 64018181, -8985428, -250991300, -727998021, -1324662165

Offset: 0

Seiichi Manyama, Aug 10 2024

sign

Cf. A375021, A375293.

Cf. A246840.

my(N=40, x='x+O('x^N)); Vec(1/sqrt((1-x+x^3)^2+4*x^4))

a(n) = sum(k=0, n\3, (-1)^k*binomial(n-2*k, k)^2);

n*a(n) = (2*n-1)*a(n-1) - (n-1)*a(n-2) - (2*n-3)*a(n-3) - 2*(n-2)*a(n-4) - (n-3)*a(n-6).

a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-2*k,k)^2.

A387508 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(n-3*k,k)^2.

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 19, 33, 55, 109, 243, 529, 1071, 2093, 4179, 8673, 18255, 37981, 77923, 159649, 329935, 687117, 1432403, 2977505, 6179215, 12841597, 26757059, 55840033, 116551119, 243209325, 507658803, 1060551137, 2217515151, 4639042909, 9707403811

Offset: 0

Seiichi Manyama, Aug 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001850, A108488, A387507.

Cf. A246883, A375293.

[(&+[2^k * Binomial(n-3*k, k)^2: k in [0..Floor(n/4)]]): n in [0..40]]; // Vincenzo Librandi, Sep 02 2025

Table[Sum[2^k*Binomial[n-3*k, k]^2,{k,0,Floor[n/4]}],{n,0,40}] (* Vincenzo Librandi, Sep 02 2025 *)

a(n) = sum(k=0, n\4, 2^k*binomial(n-3*k, k)^2);

G.f.: 1/sqrt((1-x-2*x^4)^2 - 8*x^5).

A375290 Expansion of 1/((1 - x + x^4)^2 + 4*x^5).

Original entry on oeis.org

1, 2, 3, 4, 3, -4, -21, -52, -98, -144, -143, 0, 440, 1368, 2891, 4752, 5831, 3438, -7330, -33384, -81044, -148610, -211283, -197280, 39748, 732646, 2152660, 4423184, 7089816, 8360270, 4071395, -13171888, -53480919, -125422768, -224380607, -309560644, -268524883

Offset: 0

Seiichi Manyama, Aug 10 2024

sign

Index entries for linear recurrences with constant coefficients, signature (2,-1,0,-2,-2,0,0,-1).

Cf. A375255, A375288.

Cf. A375293.

my(N=40, x='x+O('x^N)); Vec(1/((1-x+x^4)^2+4*x^5))

a(n) = sum(k=0, n\4, (-1)^k*binomial(2*n-6*k+2, 2*k+1))/2;

a(n) = 2*a(n-1) - a(n-2) - 2*a(n-4) - 2*a(n-5) - a(n-8).

a(n) = (1/2) * Sum_{k=0..floor(n/4)} (-1)^k * binomial(2*n-6*k+2,2*k+1).

cp's OEIS Frontend

A375292 Expansion of 1/sqrt((1 - x + x^3)^2 + 4*x^4).

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A387508 a(n) = Sum_{k=0..floor(n/4)} 2^k * binomial(n-3*k,k)^2.

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A375290 Expansion of 1/((1 - x + x^4)^2 + 4*x^5).

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Author

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PARI

PARI

Formula