cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A377011 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 6, 34, 188, 1026, 5556, 29940, 160824, 862018, 4613636, 24667644, 131795912, 703812916, 3757135752, 20051429544, 106992663408, 570827898306, 3045193326372, 16244056119084, 86646747723048, 462161936699196, 2465043081687192, 13147597801986264, 70123266087502608
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A293490, A383832, A386942.
Cf. A000302, A000984, A026641, A141223, A386957.

Programs

  • Magma
    [&+[3^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[3^k * Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(2*n+1, n-k));

Formula

a(n) = [x^n] 1/((1-4*x) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 4^k * binomial(2*n-k,n-k).
G.f.: 1/( sqrt(1-4*x) * (2*sqrt(1-4*x)-1) ).
a(n) ~ 2^(4*n+2) / 3^(n+1). - Vaclav Kotesovec, Aug 20 2025
D-finite with recurrence 3*n*a(n) +2*(-14*n+3)*a(n-1) +32*(2*n-1)*a(n-2)=0. - R. J. Mathar, Aug 21 2025

A386956 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+1,n-k).

Original entry on oeis.org

1, 19, 282, 3763, 47294, 571950, 6733668, 77723187, 883589238, 9924844474, 110396411372, 1218075749934, 13348677037868, 145438914042172, 1576690043132376, 17018212213758771, 182983432175308710, 1960781840268630786, 20947171352106580284, 223169444039365834362
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A088218, A258431, A384365, A386955.
Cf. A386957.

Programs

  • Magma
    [&+[(k+1) * 8^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 03 2025
  • Mathematica
    Table[Sum[(k+1) * 8^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, (k+1)*8^k*binomial(2*n+1, n-k));

Formula

a(n) = [x^n] 1/((1-9*x)^2 * (1-x)^n).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
a(n) = Sum_{k=0..n} (k+1) * 9^k * binomial(2*n-k-1,n-k).
G.f.: 2 * (1+sqrt(1-4*x))/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7)^2 ).

A387085 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(2*n+1,k).

Original entry on oeis.org

1, 0, 4, 8, 36, 120, 456, 1680, 6340, 23960, 91224, 348656, 1337896, 5149872, 19877904, 76907808, 298176516, 1158168792, 4505865144, 17555689008, 68490100536, 267518448912, 1046041377264, 4094231982048, 16039426479336, 62887835652720, 246761907761776, 968943740083040
Offset: 0

Views

Author

Seiichi Manyama, Aug 16 2025

Keywords

Links

Crossrefs

Cf. A000302, A000984, A001700, A026641, A141223, A377011, A386957.
Cf. A226733, A226705, A226751.
Cf. A005810, A079589, A183160.

Programs

  • Magma
    [&+[(-3)^(n-k) * Binomial(2*n+1,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 31 2025
  • Mathematica
    Table[Sum[(-3)^(n-k)*Binomial[2*n+1,k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 31 2025 *)
  • PARI
    a(n) = sum(k=0, n, (-3)^(n-k)*binomial(2*n+1, k));

Formula

a(n) = [x^n] (1+x)^(2*n+1)/(1+3*x).
a(n) = [x^n] 1/((1-x)^(n+1) * (1+2*x)).
a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} (-2)^k * binomial(2*n-k,n-k).
G.f.: 1/( 4*x - 1 + 2*sqrt(1 - 4*x) ).
G.f.: 1/(1 - 4*x*(-1+g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: g^2/((-2+3*g) * (2-g)) where g = 1+x*g^2 is the g.f. of A000108.
G.f.: B(x)^2/(1 + 2*(B(x)-1)), where B(x) is the g.f. of A000984.
D-finite with recurrence 3*n*a(n) +2*(-4*n+3)*a(n-1) +8*(-2*n+1)*a(n-2)=0. - R. J. Mathar, Aug 19 2025

A386958 a(n) = Sum_{k=0..n} 8^k * binomial(k-2/3,k) * binomial(2*n+1/3,n-k).

Original entry on oeis.org

1, 5, 33, 248, 2020, 17325, 153699, 1395084, 12868839, 120127865, 1131633217, 10737438816, 102480890512, 982880111192, 9465545374920, 91479218990688, 886803360846876, 8619761335490460, 83982810424366860, 819973263265010400, 8020986875021209320
Offset: 0

Views

Author

Seiichi Manyama, Aug 11 2025

Keywords

Links

Crossrefs

Cf. A386957.

Programs

  • Mathematica
    Table[Sum[8^k*Binomial[k-2/3,k]*Binomial[2*n+1/3, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Sep 03 2025 *)
  • PARI
    a(n) = sum(k=0, n, 8^k*binomial(k-2/3, k)*binomial(2*n+1/3, n-k));

Formula

a(n) = [x^n] 1/((1-9*x)^(1/3) * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+1/3,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} 9^k * binomial(k-2/3,k) * binomial(2*n-k,n-k).
G.f.: 1/( sqrt(1-4*x) * ((9*sqrt(1-4*x)-7)/2)^(1/3) ).

A386986 a(n) = Sum_{k=0..n} (k+1) * 8^k * binomial(2*n+2,n-k).

Original entry on oeis.org

1, 20, 303, 4088, 51730, 628488, 7423899, 85904688, 978506478, 11008191800, 122603713078, 1354213651728, 14854030654372, 161966063719712, 1757042561230515, 18976059641899872, 204140891541240918, 2188510439907779064, 23389705325379996834, 249285017279237071440
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2025

Keywords

Links

Crossrefs

Cf. A386957, A386958.
Cf. A000984, A002457, A383832.

Programs

  • Magma
    [&+[(k+1)*8^k * Binomial(2*n+2, n-k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 14 2025
  • Mathematica
    Table[Sum[(k+1)* 8^k*Binomial[2*n+2,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 14 2025 *)
  • PARI
    a(n) = sum(k=0, n, (k+1)*8^k*binomial(2*n+2, n-k));

Formula

a(n) = [x^n] 1/((1-9*x)^2 * (1-x)^(n+1)).
a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n+2,k) * binomial(2*n-k,n-k).
a(n) = Sum_{k=0..n} (k+1) * 9^k * binomial(2*n-k,n-k).
G.f.: 4/( sqrt(1-4*x) * (9*sqrt(1-4*x)-7)^2 ).
a(n) ~ 7 * n * 3^(4*n+2) / 2^(3*n+6). - Vaclav Kotesovec, Aug 12 2025
D-finite with recurrence 520*n*a(n) +(-8641*n-1633)*a(n-1) +486*(81*n-32)*a(n-2) +26244*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Aug 19 2025
Showing 1-5 of 5 results.

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