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 1-7 of 7 results.

A365525 a(n) = Sum_{k=0..floor(n/4)} Stirling2(n,4*k).

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 65, 350, 1702, 7806, 34855, 157630, 770529, 4432220, 31307432, 259090260, 2316320073, 21172354778, 193091210857, 1744478148866, 15627203762926, 139526376391986, 1251976261264071, 11417796498945894, 107280845105151601
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2023

Keywords

Links

Crossrefs

Cf. A024430, A143815, A365528.
Cf. A099948, A365526, A365527.

Programs

  • Maple
    f:= proc(n) local k; add(Stirling2(n,4*k),k=0..n/4) end proc:
map(f, [$0..30]); # Robert Israel, Sep 11 2024
  • Mathematica
    a[n_] := Sum[StirlingS2[n, 4*k], {k, 0, Floor[n/4]}]; Array[a, 25, 0] (* Amiram Eldar, Sep 11 2023 *)
  • PARI
    a(n) = sum(k=0, n\4, stirling(n, 4*k, 2));
  • Python
    from sympy.functions.combinatorial.numbers import stirling
def A365525(n): return sum(stirling(n,k<<2) for k in range((n>>2)+1)) # Chai Wah Wu, Sep 08 2023

Formula

Let A(0)=1, B(0)=0, C(0)=0 and D(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), D(n+1) = Sum_{k=0..n} binomial(n,k)*C(k) and A(n+1) = Sum_{k=0..n} binomial(n,k)*D(k). a(n) = A(n), A365526(n) = B(n), A365527(n) = C(n) and A099948(n) = D(n).
G.f.: Sum_{k>=0} x^(4*k) / Product_{j=1..4*k} (1-j*x).
a(n) ~ n^n / (4 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Jun 10 2025

A365529 a(n) = Sum_{k=0..floor((n-1)/5)} Stirling2(n,5*k+1).

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 2, 22, 267, 2647, 22828, 179489, 1323719, 9323744, 63502440, 422172752, 2763863468, 18017811013, 119078265944, 822495346707, 6206943675825, 53413341096271, 529613886789747, 5863983528090106, 69211078916780252, 839908976768680556
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2023

Keywords

Crossrefs

Cf. A365528, A365530, A365531, A365532.

Programs

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

Formula

Let A(0)=1, B(0)=0, C(0)=0, D(0)=0 and E(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), D(n+1) = Sum_{k=0..n} binomial(n,k)*C(k), E(n+1) = Sum_{k=0..n} binomial(n,k)*D(k) and A(n+1) = Sum_{k=0..n} binomial(n,k)*E(k). A365528(n) = A(n), a(n) = B(n), A365530(n) = C(n), A365531(n) = D(n) and A365532(n) = E(n).
G.f.: Sum_{k>=0} x^(5*k+1) / Product_{j=1..5*k+1} (1-j*x).

A365530 a(n) = Sum_{k=0..floor((n-2)/5)} Stirling2(n,5*k+2).

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 31, 64, 155, 717, 6391, 65010, 629444, 5719597, 49340838, 408864186, 3284672489, 25770192646, 198718943490, 1516391860879, 11554571944615, 89144035246500, 711587142257776, 6054854693784594, 56609279400922224, 590143167134961765
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2023

Keywords

Crossrefs

Cf. A365528, A365529, A365531, A365532.

Programs

  • PARI
    a(n) = sum(k=0, (n-2)\5, stirling(n, 5*k+2, 2));

Formula

Let A(0)=1, B(0)=0, C(0)=0, D(0)=0 and E(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), D(n+1) = Sum_{k=0..n} binomial(n,k)*C(k), E(n+1) = Sum_{k=0..n} binomial(n,k)*D(k) and A(n+1) = Sum_{k=0..n} binomial(n,k)*E(k). A365528(n) = A(n), A365529(n) = B(n), a(n) = C(n), A365531(n) = D(n) and A365532(n) = E(n).
G.f.: Sum_{k>=0} x^(5*k+2) / Product_{j=1..5*k+2} (1-j*x).

A365531 a(n) = Sum_{k=0..floor((n-3)/5)} Stirling2(n,5*k+3).

Original entry on oeis.org

0, 0, 0, 1, 6, 25, 90, 301, 967, 3061, 10080, 40381, 245553, 2161238, 21701381, 219007491, 2149071359, 20442363031, 189226358659, 1712836890912, 15232581945180, 133717667932475, 1164901223314180, 10143255631462661, 89207257764369032, 804712211338739040
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2023

Keywords

Crossrefs

Cf. A365528, A365529, A365530, A365532.

Programs

  • PARI
    a(n) = sum(k=0, (n-3)\5, stirling(n, 5*k+3, 2));

Formula

Let A(0)=1, B(0)=0, C(0)=0, D(0)=0 and E(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), D(n+1) = Sum_{k=0..n} binomial(n,k)*C(k), E(n+1) = Sum_{k=0..n} binomial(n,k)*D(k) and A(n+1) = Sum_{k=0..n} binomial(n,k)*E(k). A365528(n) = A(n), A365529(n) = B(n), A365530(n) = C(n), a(n) = D(n) and A365532(n) = E(n).
G.f.: Sum_{k>=0} x^(5*k+3) / Product_{j=1..5*k+3} (1-j*x).

A365532 a(n) = Sum_{k=0..floor((n-4)/5)} Stirling2(n,5*k+4).

Original entry on oeis.org

0, 0, 0, 0, 1, 10, 65, 350, 1701, 7771, 34150, 146905, 633776, 2892032, 15526876, 109484545, 992589171, 10223409493, 108982611518, 1156117871286, 12062817285396, 123603289559039, 1245986248828926, 12391614409960544, 121996350285087172
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2023

Keywords

Crossrefs

Cf. A365528, A365529, A365530, A365531.

Programs

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

Formula

Let A(0)=1, B(0)=0, C(0)=0, D(0)=0 and E(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), D(n+1) = Sum_{k=0..n} binomial(n,k)*C(k), E(n+1) = Sum_{k=0..n} binomial(n,k)*D(k) and A(n+1) = Sum_{k=0..n} binomial(n,k)*E(k). A365528(n) = A(n), A365529(n) = B(n), A365530(n) = C(n), A365531(n) = D(n) and a(n) = E(n).
G.f.: Sum_{k>=0} x^(5*k+4) / Product_{j=1..5*k+4} (1-j*x).

A384836 a(n) = Sum_{k=0..floor(n/4)} |Stirling1(n,4*k)|.

Original entry on oeis.org

1, 0, 0, 0, 1, 10, 85, 735, 6770, 67320, 724550, 8427650, 105615500, 1420941600, 20448793300, 313670857500, 5111631733000, 88224807112000, 1608190674259000, 30879323250633000, 623074177992110000, 13182400475167560000, 291842125111122170000, 6748135840840046510000
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2025

Keywords

Crossrefs

Cf. A000142, A001710, A357828, A384837.
Cf. A000110, A024430, A143815, A365525, A365528.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n, 4*k]], {k, 0, Floor[n/4]}], {n, 0, 30}]
  • PARI
    a(n) = sum(k=0, n\4, abs(stirling(n, 4*k, 1))); \\ Michel Marcus, Jun 10 2025

Formula

a(n) ~ n!/4.

A384837 a(n) = Sum_{k=0..floor(n/5)} |Stirling1(n,5*k)|.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 175, 1960, 22449, 269326, 3416985, 45997655, 657262606, 9959178229, 159758917956, 2707741441460, 48389066401764, 909877831207125, 17965423056654249, 371766710374672096, 8047954162682335066, 181941000229690525197, 4288430328840863166236, 105226297616943093770399
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 10 2025

Keywords

Crossrefs

Cf. A000142, A001710, A357828, A384836.
Cf. A000110, A024430, A143815, A365525, A365528.

Programs

  • Mathematica
    Table[Sum[Abs[StirlingS1[n, 5*k]], {k, 0, Floor[n/5]}], {n, 0, 30}]
  • PARI
    a(n) = sum(k=0, n\5, abs(stirling(n, 5*k, 1))); \\ Michel Marcus, Jun 10 2025

Formula

a(n) ~ n!/5.
Showing 1-7 of 7 results.

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