A298567 -id:A298567 - cp's OEIS frontend

A387483 a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(k,n-2*k)^2.

1, 0, 2, 4, 4, 32, 24, 144, 304, 576, 2336, 3648, 13120, 30208, 70528, 218368, 456448, 1360896, 3316224, 8311808, 23127040, 54812672, 151197696, 380669952, 978595840, 2613067776, 6540566528, 17464705024, 44764708864, 116183662592, 305637064704, 783627386880

Offset: 0

Seiichi Manyama, Aug 30 2025

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

Cf. A298567.

[(&+[2^(n-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-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-k)*binomial(k, n-2*k)^2);

G.f.: 1/sqrt((1-2*x^2-4*x^3)^2 - 32*x^5).

A375218 a(n) = Sum_{k=0..floor(n/2)} (k+1) * binomial(k,n-2*k)^2.

Original entry on oeis.org

1, 0, 2, 2, 3, 12, 7, 36, 41, 84, 186, 230, 612, 852, 1733, 3198, 5112, 10628, 16873, 32562, 57463, 99892, 188103, 319188, 591982, 1040076, 1849352, 3351304, 5854119, 10610416, 18707180, 33370938, 59618393, 105291208, 188572347, 333462928, 593859439, 1055432400, 1870161060

Offset: 0

Seiichi Manyama, Oct 17 2024

nonn

Cf. A182884, A375470.

Cf. A298567.

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

G.f.: (1-x^2-x^3)/((1-x^2-x^3)^2 - 4*x^5)^(3/2).

D-finite with recurrence 2*n*(2*n+1)*a(n) +3*(n-1)*(2*n-3)*a(n-1) +4*(-2*n^2-3*n+4)*a(n-2) +2*(-10*n^2+n+27)*a(n-3) +2*(-4*n^2+11*n+27)*a(n-4) +(-2*n^2-27*n-27)*a(n-5) +2*(-4*n^2+7*n+18)*a(n-6) +3*(2*n+3)*(n-1)*a(n-7)=0. - R. J. Mathar, Oct 17 2024

A376721 Expansion of 1/sqrt((1 - x^3 - x^4)^2 - 4*x^7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 4, 1, 1, 9, 9, 2, 16, 36, 17, 26, 100, 101, 61, 226, 401, 274, 477, 1227, 1289, 1225, 3186, 4982, 4432, 7841, 16040, 17902, 21457, 45517, 66610, 71327, 123444, 219825, 261945, 354095, 660573, 938598, 1138806, 1909676, 3125553

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

Cf. A051286, A298567, A376722.

Cf. A246883.

my(N=50, x='x+O('x^N)); Vec(1/sqrt((1-x^3-x^4)^2-4*x^7))

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

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

A376722 Expansion of 1/sqrt((1 - x^4 - x^5)^2 - 4*x^9).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 4, 1, 0, 1, 9, 9, 1, 1, 16, 36, 16, 2, 25, 100, 100, 26, 37, 225, 400, 226, 85, 442, 1225, 1226, 505, 833, 3137, 4901, 3217, 2080, 7120, 15878, 15976, 9081, 15696, 44182, 63626, 47125, 41625, 110926, 213688, 217801, 157300, 272251, 630458

Offset: 0

Seiichi Manyama, Oct 02 2024

nonn

Cf. A051286, A298567, A376721.

Cf. A246884.

my(N=60, x='x+O('x^N)); Vec(1/sqrt((1-x^4-x^5)^2-4*x^9))

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

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

A377146 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(k,n-2*k)^2.

Original entry on oeis.org

1, 0, 3, 3, 6, 24, 16, 90, 105, 250, 561, 765, 2143, 3108, 6861, 12985, 22221, 47988, 79463, 161451, 293610, 535836, 1042188, 1835898, 3534766, 6399198, 11805756, 22021232, 39718497, 74193924, 134489713, 247165839, 453235266, 822748406, 1512078192, 2741606052

Offset: 0

Seiichi Manyama, Oct 17 2024

nonn

Cf. A377145, A377147.
Cf. A298567, A375218.
Cf. A089627.

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

a089627(n, k) = n!/((n-2*k)!*k!^2);
my(N=2, M=40, x='x+O('x^M), X=1-x^2-x^3, Y=5); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))

G.f.: ((1-x^2-x^3)^2 + 2*x^5) / ((1-x^2-x^3)^2 - 4*x^5)^(5/2).

cp's OEIS Frontend

A387483 a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(k,n-2*k)^2.

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A375218 a(n) = Sum_{k=0..floor(n/2)} (k+1) * binomial(k,n-2*k)^2.

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A376721 Expansion of 1/sqrt((1 - x^3 - x^4)^2 - 4*x^7).

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A376722 Expansion of 1/sqrt((1 - x^4 - x^5)^2 - 4*x^9).

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A377146 a(n) = Sum_{k=0..floor(n/2)} binomial(k+2,2) * binomial(k,n-2*k)^2.

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