A357829 -id:A357829 - cp's OEIS frontend

A357828 a(n) = Sum_{k=0..floor(n/3)} |Stirling1(n,3*k)|.

1, 0, 0, 1, 6, 35, 226, 1645, 13454, 122661, 1236018, 13656951, 164290182, 2138379243, 29949509226, 449188719525, 7183702249542, 122039922034485, 2194928052851898, 41666342509646127, 832547791827455886, 17466905709043534107, 383908421683657311714

Offset: 0

Seiichi Manyama, Oct 14 2022

nonn

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Cf. A357829, A357830.
Cf. A000142, A001710, A384836, A384837.
Cf. A003703.

a(n) = sum(k=0, n\3, abs(stirling(n, 3*k, 1)));

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

Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Pochhammer(1, n)+Pochhammer(w, n)+Pochhammer(w^2, n))/3;

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(-log(1-x)).
a(n) = ( (1)_n + (w)_n + (w^2)_n )/3, where (x)_n is the Pochhammer symbol.
a(n) ~ n!/3. - Vaclav Kotesovec, Jun 10 2025

A357830 a(n) = Sum_{k=0..floor((n-2)/3)} |Stirling1(n,3*k+2)|.

Original entry on oeis.org

0, 0, 1, 3, 11, 51, 289, 1939, 15029, 132069, 1296771, 14063721, 166897059, 2150579067, 29895590361, 445871456667, 7100686041813, 120249378265653, 2157637558311963, 40887284144179473, 815949872494416387, 17103401793743095467, 375692072337527815233

Offset: 0

Seiichi Manyama, Oct 14 2022

nonn

Robert Israel, Table of n, a(n) for n = 0..448
Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Cf. A357828, A357829.

f:= proc(n) local k;  add(abs(Stirling1(n,3*k+2)), k=0..(n-2)/3) end proc:
map(f, [$0..30]); # Robert Israel, Feb 12 2024

Table[Sum[Abs[StirlingS1[n,3k+2]],{k,0,Floor[(n-2)/3]}],{n,0,30}] (* Harvey P. Dale, Jan 12 2024 *)

a(n) = sum(k=0, (n-2)\3, abs(stirling(n, 3*k+2, 1)));

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

Pochhammer(x, n) = prod(k=0, n-1, x+k);
a(n) = my(w=(-1+sqrt(3)*I)/2); round(Pochhammer(1, n)+w*Pochhammer(w, n)+w^2*Pochhammer(w^2, n))/3;

Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(-log(1-x)).

a(n) = ( (1)_n + w * (w)_n + w^2 * (w^2)_n )/3, where (x)_n is the Pochhammer symbol.

cp's OEIS Frontend

A357828 a(n) = Sum_{k=0..floor(n/3)} |Stirling1(n,3*k)|.

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A357830 a(n) = Sum_{k=0..floor((n-2)/3)} |Stirling1(n,3*k+2)|.

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