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 1-3 of 3 results.

A385670 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).

Original entry on oeis.org

1, 17, 429, 12048, 355501, 10792737, 333781044, 10457735928, 330823760061, 10543365694707, 338004221112309, 10887987584565108, 352127854740967596, 11426385227977214252, 371844089088280093224, 12130745906826301055088, 396599383187880024765981
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2025

Keywords

Crossrefs

Cf. A385669, A385671.
Cf. A386767.

Programs

  • PARI
    a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(3*n+1, k)*binomial(3*n-k, n-k));

Formula

a(n) = [x^n] (1+2*x)^(3*n+1)/(1-3*x)^(2*n+1).
a(n) = [x^n] 1/((1-2*x) * (1-5*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 5^k * (-3)^(n-k) * binomial(3*n+1,k).
a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(2*n+k,k).

A385671 a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(4*n+1,k) * binomial(4*n-k,n-k).

Original entry on oeis.org

1, 22, 774, 30458, 1260886, 53731512, 2333065354, 102643195068, 4559878830006, 204091261040552, 9189096061165784, 415734554486178378, 18884084064916032026, 860673634902720476392, 39339618388269633525564, 1802605962076744803396888, 82777622289467318635747446
Offset: 0

Views

Author

Seiichi Manyama, Aug 04 2025

Keywords

Crossrefs

Cf. A016127, A385669, A385670.
Cf. A384366, A386768.

Programs

  • PARI
    a(n) = sum(k=0, n, 2^k*3^(n-k)*binomial(4*n+1, k)*binomial(4*n-k, n-k));

Formula

a(n) = [x^n] (1+2*x)^(4*n+1)/(1-3*x)^(3*n+1).
a(n) = [x^n] 1/((1-2*x) * (1-5*x)^(3*n+1)).
a(n) = Sum_{k=0..n} 5^k * (-3)^(n-k) * binomial(4*n+1,k).
a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(3*n+k,k).

A386829 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(2*n+1,k) * binomial(2*n-k,n-k).

Original entry on oeis.org

1, 13, 204, 3457, 61006, 1103598, 20299434, 377871297, 7097430726, 134243202358, 2553356761264, 48788507855562, 935791802540596, 18007015501848568, 347459946354962694, 6720599552926105377, 130263082422599127366, 2529516572366126192478
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2025

Keywords

Crossrefs

Cf. A386830, A386831.
Cf. A385669, A386763.

Programs

  • PARI
    a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(2*n+1, k)*binomial(2*n-k, n-k));

Formula

a(n) = [x^n] (1+3*x)^(2*n+1)/(1-2*x)^(n+1).
a(n) = [x^n] 1/((1-3*x) * (1-5*x)^(n+1)).
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(2*n+1,k).
a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(n+k,k).
Showing 1-3 of 3 results.

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

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