A375276 -id:A375276 - cp's OEIS frontend

A387476 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(k,n-2*k)^2.

1, 0, 2, 2, 4, 16, 12, 72, 88, 264, 608, 1056, 3280, 5504, 15328, 31904, 71104, 175488, 358080, 900736, 1925248, 4518016, 10404864, 23138304, 54970624, 122038272, 286077440, 651510272, 1492685824, 3465687040, 7876488192, 18322630656, 41904609280, 96788580352, 223335882752

Offset: 0

Seiichi Manyama, Aug 30 2025

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

Cf. A375276, A387477.

Cf. A052907.

[(&+[2^k * Binomial(k,n-2*k)^2: k in [0..Floor(n/2)]]): n in [0..40]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[2^k* Binomial[k,n-2*k]^2,{k,0,Floor[n/2]}],{n,0,40}] (* Vincenzo Librandi, Aug 31 2025 *)

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

G.f.: 1/sqrt((1-2*x^2-2*x^3)^2 - 16*x^5).

A387477 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(k,n-3*k)^2.

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 4, 16, 4, 8, 72, 72, 24, 256, 576, 288, 816, 3200, 3264, 3104, 14432, 25728, 20672, 58752, 157120, 173184, 257152, 809600, 1296000, 1466368, 3814400, 8247296, 10202368, 18360320, 46069760, 71264768, 100919808, 238362624, 457049088, 635490304

Offset: 0

Seiichi Manyama, Aug 30 2025

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

Cf. A375276, A387476.

[(&+[2^k * Binomial(k,n-3*k)^2: k in [0..Floor(n/3)]]): n in [0..40]]; // Vincenzo Librandi, Aug 31 2025

Table[Sum[2^k*Binomial[k,n-3*k]^2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Aug 31 2025 *)

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

G.f.: 1/sqrt((1-2*x^3-2*x^4)^2 - 16*x^7).

A387509 a(n) = Sum_{k=0..floor(n/3)} 2^(n-k) * binomial(n-2*k,k)^2.

Original entry on oeis.org

1, 2, 4, 12, 48, 176, 592, 2016, 7168, 25920, 93440, 336384, 1216768, 4425216, 16143360, 58993664, 215965696, 792154112, 2910720000, 10710884352, 39464009728, 145575002112, 537583419392, 1987177611264, 7352243781632, 27224715689984, 100888181211136

Offset: 0

Seiichi Manyama, Aug 31 2025

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

Cf. A375276.

[(&+[2^(n-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^(n-k)*Binomial[n-2*k,k]^2,{k,0,Floor[n/3]}],{n,0,30}] (* Vincenzo Librandi, Sep 02 2025 *)

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

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

A387550 a(n) = (1/2) * Sum_{k=0..n} 2^k * binomial(2k+2,2n-2*k+1).

Original entry on oeis.org

1, 4, 16, 72, 316, 1376, 6016, 26304, 114960, 502464, 2196224, 9599360, 41957312, 183389184, 801566720, 3503527936, 15313395968, 66932560896, 292552200192, 1278701856768, 5589014330368, 24428744679424, 106774384771072, 466694846300160, 2039853285314560

Offset: 0

Seiichi Manyama, Sep 01 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,8,-4).

Cf. A375276.

[&+[2^k * Binomial(2*k+2, 2*n-2*k+1)/2: k in [0..n]]: n in [0..40]]; // Vincenzo Librandi, Sep 02 2025

Table[Sum[2^k*Binomial[2*k+2,2*n-2*k+1]/2,{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Sep 02 2025 *)

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

G.f.: B(x)^2, where B(x) is the g.f. of A375276.
G.f.: 1/((1-2*x-2*x^2)^2 - 16*x^3).
a(n) = 4*a(n-1) + 8*a(n-3) - 4*a(n-4).

cp's OEIS Frontend

A387476 a(n) = Sum_{k=0..floor(n/2)} 2^k * binomial(k,n-2*k)^2.

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A387477 a(n) = Sum_{k=0..floor(n/3)} 2^k * binomial(k,n-3*k)^2.

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Author

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Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387509 a(n) = Sum_{k=0..floor(n/3)} 2^(n-k) * binomial(n-2*k,k)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387550 a(n) = (1/2) * Sum_{k=0..n} 2^k * binomial(2*k+2,2*n-2*k+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387550 a(n) = (1/2) * Sum_{k=0..n} 2^k * binomial(2k+2,2n-2*k+1).