A108490 -id:A108490 - cp's OEIS frontend

A387480 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(k,n-k)^2.

1, 3, 15, 99, 603, 3807, 24759, 162243, 1072683, 7147359, 47887767, 322330995, 2178055899, 14765637663, 100380161655, 684061007139, 4671543976587, 31962145170015, 219043736154711, 1503380943222867, 10332034575214779, 71092843087100319, 489712662842798007

Offset: 0

Seiichi Manyama, Aug 30 2025

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

Cf. A387481, A387482.

Cf. A083858, A108490.

[&+[3^k * 2^(n-k) * Binomial(k, n-k)^2: k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[3^k*2^(n-k)*Binomial[k,n-k]^2,{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Sep 01 2025 *)

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

G.f.: 1/sqrt((1-3*x-6*x^2)^2 - 72*x^3).

A387478 a(n) = Sum_{k=0..floor(n/2)} 2^k * 3^(n-2k) binomial(k,n-2*k)^2.

Original entry on oeis.org

1, 0, 2, 6, 4, 48, 44, 216, 664, 984, 5216, 9312, 30160, 93312, 194528, 717792, 1674688, 4842624, 14554304, 35517312, 112151680, 293213568, 823387136, 2409934848, 6348908800, 18760218624, 51418362368, 143838905856, 414017065984, 1132211048448, 3255687793664

Offset: 0

Seiichi Manyama, Aug 30 2025

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

Cf. A108490, A387479.

[(&+[2^k *3^(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^k* 3^(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^k*3^(n-2*k)*binomial(k, n-2*k)^2);

G.f.: 1/sqrt((1-2*x^2-6*x^3)^2 - 48*x^5).

A387479 a(n) = Sum_{k=0..floor(n/3)} 2^k * 3^(n-3k) binomial(k,n-3*k)^2.

Original entry on oeis.org

1, 0, 0, 2, 6, 0, 4, 48, 36, 8, 216, 648, 232, 768, 5184, 6944, 3696, 28800, 86464, 71712, 137376, 691328, 1185216, 1067904, 4280512, 12749952, 15523200, 26248832, 102010752, 201056256, 243856384, 694548480, 1995570432, 3031771136, 5109762048, 16129681920

Offset: 0

Seiichi Manyama, Aug 30 2025

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

Cf. A108490, A387478.

[(&+[2^k *3^(n-3*k)* Binomial(k,n-3*k)^2: k in [0..Floor(n/3)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[2^k* 3^(n-3*k)*Binomial[k,n-3*k]^2,{k,0,Floor[n/3]}],{n,0,40}] (* Vincenzo Librandi, Sep 01 2025 *)

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

G.f.: 1/sqrt((1-2*x^3-6*x^4)^2 - 48*x^7).

A387511 a(n) = Sum_{k=0..floor(n/3)} 3^k * 2^(n-2k) binomial(n-2*k,k)^2.

Original entry on oeis.org

1, 2, 4, 14, 64, 248, 868, 3176, 12352, 48344, 186688, 720896, 2810128, 11021984, 43290688, 170193632, 670576384, 2648370560, 10477291072, 41502538880, 164602863616, 653632824704, 2598446927872, 10339935936512, 41181966803200, 164155849556480, 654848284582912

Offset: 0

Seiichi Manyama, Aug 31 2025

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

Cf. A108490.

[(&+[3^k * 2^(n-2*k) * Binomial(n-2*k, k)^2: k in [0..Floor(n/3)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025

Table[Sum[3^k*2^(n-2*k)*Binomial[n-2*k,k]^2,{k,0,Floor[n/3]}],{n,0,30}] (* Vincenzo Librandi, Sep 01 2025 *)

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

G.f.: 1/sqrt((1-2*x-6*x^3)^2 - 48*x^4).

cp's OEIS Frontend

A387480 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(k,n-k)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387478 a(n) = Sum_{k=0..floor(n/2)} 2^k * 3^(n-2*k) * binomial(k,n-2*k)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387479 a(n) = Sum_{k=0..floor(n/3)} 2^k * 3^(n-3*k) * binomial(k,n-3*k)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387511 a(n) = Sum_{k=0..floor(n/3)} 3^k * 2^(n-2*k) * binomial(n-2*k,k)^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A387478 a(n) = Sum_{k=0..floor(n/2)} 2^k * 3^(n-2k) binomial(k,n-2*k)^2.

A387479 a(n) = Sum_{k=0..floor(n/3)} 2^k * 3^(n-3k) binomial(k,n-3*k)^2.

A387511 a(n) = Sum_{k=0..floor(n/3)} 3^k * 2^(n-2k) binomial(n-2*k,k)^2.