A108488 -id:A108488 - cp's OEIS frontend

A375021 Expansion of 1/sqrt(1 - 2x + 3x^2 + 2*x^3 + x^4).

1, 1, 0, -3, -7, -6, 11, 49, 78, 3, -297, -750, -691, 1271, 5970, 9877, 647, -38640, -100381, -95689, 170394, 827453, 1398933, 131418, -5472241, -14495327, -14186826, 24241947, 121177521, 208360152, 25541493, -807963639, -2175698844, -2179521039

Offset: 0

Seiichi Manyama, Aug 09 2024

sign

Cf. A051286, A108488, A108489.

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

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

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

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

A387507 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(n-2*k,k)^2.

Original entry on oeis.org

1, 1, 1, 3, 9, 19, 37, 87, 217, 507, 1157, 2727, 6553, 15627, 37077, 88519, 212569, 510715, 1226853, 2952615, 7120921, 17192427, 41538293, 100458759, 243211865, 589313755, 1428931333, 3467193191, 8418640793, 20453853003, 49722339861, 120936710471

Offset: 0

Seiichi Manyama, Aug 31 2025

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

Cf. A001850, A108488, A387508.

Cf. A246840, A375292.

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

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

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

G.f.: 1/sqrt((1-x-2*x^3)^2 - 8*x^4).

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).

A387515 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2k) binomial(k,n-2*k)^2.

Original entry on oeis.org

1, 0, 1, 2, 1, 8, 5, 18, 37, 40, 145, 178, 417, 872, 1301, 3330, 5365, 11080, 22801, 39362, 86721, 157128, 312293, 631666, 1169541, 2416104, 4602961, 9061458, 18123553, 34717608, 69825013, 135902818, 267384405, 531611656, 1035512785, 2060791650, 4048647489, 7979180296

Offset: 0

Seiichi Manyama, Sep 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A108488, A387516.

Cf. A387481, A387483.

[(&+[2^(n-2*k) * Binomial(k,n-2*k)^2: k in [0..Floor(n/2)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[2^(n-2*k)*Binomial[k,n-2*k]^2,{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Sep 01 2025 *)

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

G.f.: 1/sqrt((1-x^2-2*x^3)^2 - 8*x^5).

A375273 Expansion of 1/(1 - 2x - 3x^2 - 4x^3 + 4x^4).

Original entry on oeis.org

1, 2, 7, 24, 73, 238, 763, 2436, 7821, 25050, 80255, 257200, 824081, 2640582, 8461187, 27111644, 86872853, 278363058, 891946503, 2858027016, 9157854361, 29344123550, 94026132235, 301283944500, 965391362461, 3093362593162, 9911930522767, 31760378496864

Offset: 0

Seiichi Manyama, Aug 09 2024

nonn

Index entries for linear recurrences with constant coefficients, signature (2,3,4,-4).

Cf. A182890, A375255.

Cf. A108480, A108488.

my(N=30, x='x+O('x^N)); Vec(1/(1-2*x-3*x^2-4*x^3+4*x^4))

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

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

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

cp's OEIS Frontend

A375021 Expansion of 1/sqrt(1 - 2*x + 3*x^2 + 2*x^3 + x^4).

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A387507 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(n-2*k,k)^2.

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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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Magma

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Formula

A387515 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2*k) * binomial(k,n-2*k)^2.

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Magma

Mathematica

PARI

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A375273 Expansion of 1/(1 - 2*x - 3*x^2 - 4*x^3 + 4*x^4).

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PARI

PARI

Formula

A375021 Expansion of 1/sqrt(1 - 2x + 3x^2 + 2*x^3 + x^4).

A387515 a(n) = Sum_{k=0..floor(n/2)} 2^(n-2k) binomial(k,n-2*k)^2.

A375273 Expansion of 1/(1 - 2x - 3x^2 - 4x^3 + 4x^4).