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.

A365699 G.f. satisfies A(x) = 1 + x^5*A(x)^2 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 3, 6, 10, 15, 21, 33, 57, 101, 175, 291, 477, 791, 1341, 2310, 3986, 6839, 11681, 19966, 34300, 59245, 102647, 177963, 308483, 534973, 929147, 1616981, 2818967, 4920299, 8594665, 15023561, 26283971, 46030771, 80695333, 141593087
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A212364, A365698, A365700, A365701, A365702.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k-1, n-5*k)*binomial(n-3*k+1, k)/(n-3*k+1));

Formula

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k-1,n-5*k) * binomial(n-3*k+1,k) / (n-3*k+1).

A365701 G.f. satisfies A(x) = 1 + x^5*A(x)^4 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 5, 10, 16, 23, 31, 62, 128, 243, 423, 686, 1192, 2223, 4223, 7843, 13991, 24856, 45108, 83673, 156223, 288535, 527971, 966803, 1784663, 3319988, 6183424, 11483613, 21284475, 39499855, 73558147, 137347615, 256616567, 479231240
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A212364, A365698, A365699, A365700, A365702.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k-1, n-5*k)*binomial(n-k+1, k)/(n-k+1));

Formula

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k-1,n-5*k) * binomial(n-k+1,k) / (n-k+1).

A365702 G.f. satisfies A(x) = 1 + x^5*A(x)^5 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 6, 12, 19, 27, 36, 81, 177, 341, 592, 951, 1726, 3417, 6766, 12812, 22951, 41531, 78222, 151291, 291957, 550015, 1024683, 1924543, 3671017, 7063893, 13532120, 25730347, 48840523, 93154161, 178806493, 343926597, 660308308, 1265195467
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A212364, A365698, A365699, A365700, A365701.
Cf. A000108, A005043, A114997, A215341.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k-1, n-5*k)*binomial(n+1, k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/5)} binomial(n-4*k-1,n-5*k) * binomial(n+1,k).

A365698 G.f. satisfies A(x) = 1 + x^5 / (1 - x*A(x)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 22, 31, 47, 76, 126, 207, 331, 517, 801, 1251, 1987, 3206, 5212, 8465, 13677, 21997, 35341, 56937, 92169, 149860, 244274, 398383, 649379, 1058055, 1724575, 2814475, 4600923, 7533150, 12347908, 20252837, 33230545
Offset: 0

Views

Author

Seiichi Manyama, Sep 16 2023

Keywords

Crossrefs

Cf. A212364, A365699, A365700, A365701, A365702.
Cf. A001006, A023426, A025246, A357308.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k-1, n-5*k)*binomial(n-5*k+1, k)/(n-5*k+1));

Formula

G.f.: A(x) = 2*(1+x^5) / (1+x+sqrt( (1+x)^2 - 4*x*(1+x^5) )).
a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k-1,n-5*k) * binomial(n-5*k+1,k) / (n-5*k+1).

A365735 G.f. satisfies A(x) = 1 + x*A(x) / (1 - x^5*A(x)^3).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 6, 16, 36, 71, 128, 223, 403, 796, 1706, 3775, 8252, 17485, 35986, 72988, 148594, 307833, 650947, 1395846, 3004732, 6443836, 13732127, 29134320, 61792707, 131525272, 281463507, 605273669, 1305373379, 2817407854, 6077804871, 13103021422
Offset: 0

Views

Author

Seiichi Manyama, Sep 17 2023

Keywords

Crossrefs

Cf. A212364, A212385, A365734, A365736.
Cf. A365733, A365700.

Programs

  • PARI
    a(n) = sum(k=0, n\5, binomial(n-4*k-1, k)*binomial(n-2*k+1, n-5*k)/(n-2*k+1));

Formula

a(n) = Sum_{k=0..floor(n/5)} binomial(n-4*k-1,k) * binomial(n-2*k+1,n-5*k) / (n-2*k+1).
Showing 1-5 of 5 results.

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