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 1-4 of 4 results.

A387339 a(n) = Sum_{k=0..n} 3^k * binomial(n+2,k) * binomial(n+2,k+2).

Original entry on oeis.org

1, 12, 108, 880, 6855, 52164, 391720, 2918304, 21634290, 159880600, 1179180552, 8685874080, 63930198787, 470327654580, 3459353475600, 25442360389696, 187126561024686, 1376455855989672, 10126540146288520, 74515694338112160, 548444877468906726
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2025

Keywords

Links

Crossrefs

Cf. A069835, A331792, A387340.
Cf. A387310.

Programs

  • Magma
    [&+[3^k * Binomial(n+2,k) * Binomial(n+2,k+2): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
  • Mathematica
    Table[Sum[3^k * Binomial[n+2,k]*Binomial[n+2, k+2],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(n+2, k)*binomial(n+2, k+2));

Formula

n*(n+4)*a(n) = (n+2) * (4*(2*n+3)*a(n-1) - 4*(n+1)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^k * 4^(n-2*k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = [x^n] (1+4*x+3*x^2)^(n+2).
E.g.f.: exp(4*x) * BesselI(2, 2*sqrt(3)*x) / 3, with offset 2.

A387311 a(n) = Sum_{k=0..n} 3^k * binomial(n+3,k+3) * binomial(2*k+6,k+6).

Original entry on oeis.org

1, 28, 535, 8750, 132041, 1900808, 26557986, 363716220, 4912064355, 65673861484, 871539802276, 11501122783696, 151118588963615, 1978948331160080, 25846338449608184, 336857447941007280, 4382848524348689883, 56947000383926523780, 739095412895790074215, 9583718189242229830798
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2025

Keywords

Links

Crossrefs

Cf. A387309, A387310.

Programs

  • Magma
    [&+[3^k * Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 29 2025
  • Mathematica
    Table[Sum[3^k * Binomial[n+3,k+3]*Binomial[2*k+6, k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 29 2025 *)
  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(n+3, k+3)*binomial(2*k+6, k+6));

Formula

n*(n+6)*a(n) = (n+3) * (7*(2*n+5)*a(n-1) - 13*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+7*x+9*x^2)^(n+3).
E.g.f.: exp(7*x) * BesselI(3, 6*x) / 27, with offset 3.

A387309 a(n) = Sum_{k=0..n} 3^k * binomial(n+1,k+1) * binomial(2*k+2,k+2).

Original entry on oeis.org

1, 14, 174, 2128, 26045, 320082, 3951493, 48987848, 609592347, 7610525650, 95287524332, 1196054790168, 15046318739803, 189654839753750, 2394743468261190, 30285593026553536, 383554551776056139, 4863775493104574634, 61748210178809072722, 784757334938247965840, 9983152795673915802399
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2025

Keywords

Links

Crossrefs

Cf. A387310, A387311.
Cf. A026376, A331793.
Cf. A385716.

Programs

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

Formula

G.f.: ((1-7*x)/sqrt((1-x) * (1-13*x)) - 1)/(18*x^2).
n*(n+2)*a(n) = (n+1) * (7*(2*n+1)*a(n-1) - 13*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 9^k * 7^(n-2*k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+7*x+9*x^2)^(n+1).
E.g.f.: exp(7*x) * BesselI(1, 6*x) / 3, with offset 1.

A387315 Expansion of 1/((1-x) * (1-13*x))^(5/2).

Original entry on oeis.org

1, 35, 825, 16415, 297220, 5067972, 82893720, 1315073760, 20381376015, 310101196405, 4648184007467, 68817616687365, 1008344472704660, 14644604899082620, 211073938188085620, 3022082811670829676, 43017189132931007655, 609159438493806780405, 8586490781973282553375
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2025

Keywords

Links

Crossrefs

Cf. A385716, A387316.
Cf. A387230.

Programs

  • Magma
    R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-13*x))^(5/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 28 2025
  • Mathematica
    CoefficientList[Series[1/((1-x)*(1-13*x))^(5/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 28 2025 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(1/((1-x)*(1-13*x))^(5/2))

Formula

n*a(n) = (14*n+21)*a(n-1) - 13*(n+3)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 13^k * binomial(-5/2,k) * binomial(-5/2,n-k).
a(n) = Sum_{k=0..n} (-12)^k * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = Sum_{k=0..n} 12^k * 13^(n-k) * binomial(-5/2,k) * binomial(n+4,n-k).
a(n) = (binomial(n+4,2)/6) * A387310(n).
a(n) = (-1)^n * Sum_{k=0..n} 14^k * (13/14)^(n-k) * binomial(-5/2,k) * binomial(k,n-k).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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