A375255 -id:A375255 - cp's OEIS frontend

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

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).

A375275 Expansion of (1 - x + x^2)/(1 - 2x + 3x^2 + 2*x^3 + x^4).

Original entry on oeis.org

1, 1, 0, -5, -13, -12, 25, 117, 196, 3, -841, -2200, -2079, 4121, 19720, 33435, 1547, -140772, -372775, -359763, 678796, 3323203, 5702319, 437200, -23557759, -63154959, -62213360, 111716475, 559940707, 972313668, 103585625, -3941367643, -10698060204

Offset: 0

Seiichi Manyama, Aug 09 2024

sign

Index entries for linear recurrences with constant coefficients, signature (2,-3,-2,-1).

Cf. A108479, A108480, A108484.

Cf. A375255.

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

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

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

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

A375288 Expansion of 1/((1 - x + x^3)^2 + 4*x^4).

Original entry on oeis.org

1, 2, 3, 2, -5, -22, -50, -74, -47, 122, 544, 1230, 1816, 1144, -3029, -13416, -30267, -44578, -27815, 75170, 330874, 744780, 1094243, 676196, -1865344, -8160100, -18326608, -26859600, -16435947, 46284926, 201243559, 450953386, 659291863, 399432970, -1148383866

Offset: 0

Seiichi Manyama, Aug 10 2024

sign

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

Cf. A375255, A375290.

Cf. A375278, A375292.

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

a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-4*k+2, 2*k+1))/2;

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

a(n) = (1/2) * Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-4*k+2,2*k+1).

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

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

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A375275 Expansion of (1 - x + x^2)/(1 - 2*x + 3*x^2 + 2*x^3 + x^4).

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A375288 Expansion of 1/((1 - x + x^3)^2 + 4*x^4).

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PARI

PARI

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

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PARI

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

A375275 Expansion of (1 - x + x^2)/(1 - 2x + 3x^2 + 2*x^3 + x^4).