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.

A365528 a(n) = Sum_{k=0..floor(n/5)} Stirling2(n,5*k).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42526, 246785, 1381105, 7547826, 40827787, 223429571, 1289945660, 8411093621, 66070626548, 624900235273, 6667243384356, 74991482322466, 854627237256694, 9698297591786441, 108934902927646609
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2023

Keywords

Crossrefs

Cf. A365529, A365530, A365531, A365532.
Cf. A024430, A143815, A365525.

Programs

  • Mathematica
    a[n_] := Sum[StirlingS2[n, 5*k], {k, 0, Floor[n/5]}]; Array[a, 25, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, n\5, stirling(n, 5*k, 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). a(n) = A(n), A365529(n) = B(n), A365530(n) = C(n), A365531(n) = D(n) and A365532(n) = E(n).
G.f.: Sum_{k>=0} x^(5*k) / Product_{j=1..5*k} (1-j*x).
a(n) ~ n^n / (5 * (LambertW(n))^n * exp(n+1-n/LambertW(n)) * sqrt(1+LambertW(n))). - Vaclav Kotesovec, Jun 10 2025

A365526 a(n) = Sum_{k=0..floor((n-1)/4)} Stirling2(n,4*k+1).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 16, 141, 1051, 6953, 42571, 247886, 1401676, 7868005, 45210257, 277899961, 1917140421, 15186484134, 135259346092, 1295096363273, 12821558136891, 128268683204737, 1283599391456735, 12817818177339530, 127998022119881272
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2023

Keywords

Links

Crossrefs

Cf. A099948, A365525, A365527.

Programs

  • Maple
    f:= proc(n) local k; add(Stirling2(n,4*k+1),k=0..(n-1)/4) end proc:
map(f, [$0..30]); # Robert Israel, Sep 11 2024
  • Mathematica
    a[n_] := Sum[StirlingS2[n, 4*k+1], {k, 0, Floor[(n-1)/4]}]; Array[a, 25, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, (n-1)\4, stirling(n, 4*k+1, 2));

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). A365525(n) = A(n), a(n) = B(n), A365527(n) = C(n) and A099948(n) = D(n).
G.f.: Sum_{k>=0} x^(4*k+1) / Product_{j=1..4*k+1} (1-j*x).

A365527 a(n) = Sum_{k=0..floor((n-2)/4)} Stirling2(n,4*k+2).

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 32, 84, 393, 2901, 23339, 180565, 1327404, 9364732, 64197317, 433372411, 2928720335, 20264399483, 147807954692, 1170622475408, 10229966924581, 97922117830589, 1001744359476291, 10661002700183905, 115706501336004984
Offset: 0

Views

Author

Seiichi Manyama, Sep 08 2023

Keywords

Crossrefs

Cf. A099948, A365525, A365526.

Programs

  • Mathematica
    a[n_] := Sum[StirlingS2[n, 4*k+2], {k, 0, Floor[(n-2)/4]}]; Array[a, 25, 0] (* Amiram Eldar, Sep 13 2023 *)
  • PARI
    a(n) = sum(k=0, (n-2)\4, stirling(n, 4*k+2, 2));

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). A365525(n) = A(n), A365526(n) = B(n), a(n) = C(n) and A099948(n) = D(n).
G.f.: Sum_{k>=0} x^(4*k+2) / Product_{j=1..4*k+2} (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 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.