A099948 -id:A099948 - cp's OEIS frontend

A358499 a(n) = Sum_{k=0..floor(n/4)} (n-4*k)!.

1, 1, 2, 6, 25, 121, 722, 5046, 40345, 363001, 3629522, 39921846, 479041945, 6227383801, 87181920722, 1307714289846, 20923268929945, 355693655479801, 6402460887648722, 121646408123121846, 2432922931445569945, 51091297865364919801, 1124007130238495328722

Offset: 0

Seiichi Manyama, Nov 19 2022

Seiichi Manyama, Table of n, a(n) for n = 0..449

Cf. A136580, A358498, A358500.

PARI
```
a(n) = sum(k=0, n\4, (n-4*k)!);
```

a(n) = n * a(n-1) + a(n-4) - n * a(n-5) for n > 4.

a(n) ~ n! * (1 + 1/n^4 + 6/n^5 + 25/n^6 + 90/n^7 + 302/n^8 + 994/n^9 + 3487/n^10 + ...), for coefficients see A099948. - Vaclav Kotesovec, Nov 24 2022

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

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

Seiichi Manyama, Sep 08 2023

nonn

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

Cf. A099948, A365525, A365527.

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

a[n_] := Sum[StirlingS2[n, 4*k+1], {k, 0, Floor[(n-1)/4]}]; Array[a, 25, 0] (* Amiram Eldar, Sep 13 2023 *)

a(n) = sum(k=0, (n-1)\4, stirling(n, 4*k+1, 2));

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

Seiichi Manyama, Sep 08 2023

nonn

Cf. A099948, A365525, A365526.

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

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

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).

cp's OEIS Frontend

A358499 a(n) = Sum_{k=0..floor(n/4)} (n-4*k)!.

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

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

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

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