A143815 -id:A143815 - cp's OEIS frontend

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

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

Seiichi Manyama, Sep 08 2023

nonn

Cf. A365529, A365530, A365531, A365532.

Cf. A024430, A143815, A365525.

a[n_] := Sum[StirlingS2[n, 5*k], {k, 0, Floor[n/5]}]; Array[a, 25, 0] (* Amiram Eldar, Sep 13 2023 *)

a(n) = sum(k=0, n\5, stirling(n, 5*k, 2));

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

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

Seiichi Manyama, Sep 08 2023

nonn

Robert Israel, Table of n, a(n) for n = 0..574

Cf. A024430, A143815, A365528.

Cf. A099948, A365526, A365527.

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

a[n_] := Sum[StirlingS2[n, 4*k], {k, 0, Floor[n/4]}]; Array[a, 25, 0] (* Amiram Eldar, Sep 11 2023 *)

a(n) = sum(k=0, n\4, stirling(n, 4*k, 2));

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

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

A357293 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} Stirling2(n,k*j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 5, 0, 1, 0, 0, 3, 15, 0, 1, 0, 0, 1, 8, 52, 0, 1, 0, 0, 0, 6, 25, 203, 0, 1, 0, 0, 0, 1, 25, 97, 877, 0, 1, 0, 0, 0, 0, 10, 91, 434, 4140, 0, 1, 0, 0, 0, 0, 1, 65, 322, 2095, 21147, 0, 1, 0, 0, 0, 0, 0, 15, 350, 1232, 10707, 115975, 0, 1, 0, 0, 0, 0, 0, 1, 140, 1702, 5672, 58194, 678570, 0

Offset: 0

Seiichi Manyama, Oct 17 2022

			Square array begins:
  1,   1,  1,  1,  1,  1, 1, ...
  0,   1,  0,  0,  0,  0, 0, ...
  0,   2,  1,  0,  0,  0, 0, ...
  0,   5,  3,  1,  0,  0, 0, ...
  0,  15,  8,  6,  1,  0, 0, ...
  0,  52, 25, 25, 10,  1, 0, ...
  0, 203, 97, 91, 65, 15, 1, ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
Eric Weisstein's World of Mathematics, Bell Polynomial.

Columns k=0-3 give: A000007, A000110, A024430, A143815.

Cf. A357119.

T(n, k) = sum(j=0, n, stirling(n, k*j, 2));

T(n, k) = if(k==0, 0^n, n!*polcoef(sum(j=0, n\k, (exp(x+x*O(x^n))-1)^(k*j)/(k*j)!), n));

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
T(n, k) = if(k==0, 0^n, my(w=exp(2*Pi*I/k)); round(sum(j=0, k-1, Bell_poly(n, w^j)))/k);

For k > 0, e.g.f. of column k: Sum_{j>=0} (exp(x)-1)^(k*j)/(k*j)!.

For k > 0, T(n,k) = ( Sum_{j=0..k-1} Bell_n(w^j) )/k, where Bell_n(x) is n-th Bell polynomial and w = exp(2*Pi*i/k).

A357782 a(n) = Sum_{k=0..floor(n/3)} 2^k * Stirling2(n,3*k).

Original entry on oeis.org

1, 0, 0, 2, 12, 50, 184, 686, 2996, 16642, 110328, 784190, 5645876, 40685762, 296458344, 2226254766, 17564381332, 147289101090, 1312394060536, 12305546886398, 119906479624084, 1202273551045474, 12341175064817576, 129582557972751918, 1394497073432776756

Offset: 0

Seiichi Manyama, Oct 13 2022

nonn

Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A357783, A357784.

Cf. A143815, A357831.

a(n) = sum(k=0, n\3, 2^k*stirling(n, 3*k, 2));

my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, 2^k*(exp(x)-1)^(3*k)/(3*k)!)))

Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, v)+Bell_poly(n, v*w)+Bell_poly(n, v*w^2))/3;

Let A(0)=1, B(0)=0 and C(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) and A(n+1) = 2 * Sum_{k=0..n} binomial(n,k)*C(k). a(n) = A(n), A357783(n) = B(n) and A357784(n) = C(n).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(2^(1/3) * (exp(x)-1)).
a(n) = ( Bell_n(2^(1/3)) + Bell_n(2^(1/3)*w) + Bell_n(2^(1/3)*w^2) )/3, where Bell_n(x) is n-th Bell polynomial.

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

Vaclav Kotesovec, Jun 10 2025

nonn

Cf. A000142, A001710, A357828, A384837.

Cf. A000110, A024430, A143815, A365525, A365528.

Table[Sum[Abs[StirlingS1[n, 4*k]], {k, 0, Floor[n/4]}], {n, 0, 30}]

a(n) = sum(k=0, n\4, abs(stirling(n, 4*k, 1))); \\ Michel Marcus, Jun 10 2025

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

Vaclav Kotesovec, Jun 10 2025

nonn

Cf. A000142, A001710, A357828, A384836.

Cf. A000110, A024430, A143815, A365525, A365528.

Table[Sum[Abs[StirlingS1[n, 5*k]], {k, 0, Floor[n/5]}], {n, 0, 30}]

a(n) = sum(k=0, n\5, abs(stirling(n, 5*k, 1))); \\ Michel Marcus, Jun 10 2025

a(n) ~ n!/5.

cp's OEIS Frontend

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

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

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A357293 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} Stirling2(n,k*j).

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A357782 a(n) = Sum_{k=0..floor(n/3)} 2^k * Stirling2(n,3*k).

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

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

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