A357832 -id:A357832 - cp's OEIS frontend

A357831 a(n) = Sum_{k=0..floor(n/3)} 2^k * |Stirling1(n,3*k)|.

1, 0, 0, 2, 12, 70, 454, 3332, 27552, 254400, 2598852, 29125932, 355455468, 4693396656, 66671326176, 1013916648840, 16436063079552, 282920894841096, 5153797995148296, 99052313167391760, 2003040751641857856, 42513854724369719136, 944959706480298199824

Offset: 0

Seiichi Manyama, Oct 14 2022

nonn

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Cf. A357832, A357833.

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

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

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

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

A357833 a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * |Stirling1(n,3*k+2)|.

Original entry on oeis.org

0, 0, 1, 3, 11, 52, 304, 2114, 16992, 154626, 1568706, 17535108, 213965520, 2828584824, 40259041188, 613656673476, 9971942784132, 172071391424832, 3141974627361216, 60523400730707208, 1226519845766281008, 26084378634267048984, 580854626450078463000

Offset: 0

Seiichi Manyama, Oct 14 2022

nonn

Eric Weisstein's World of Mathematics, Pochhammer Symbol.

Cf. A357831, A357832.

a(n) = sum(k=0, (n-2)\3, 2^k*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, 2^k*(-log(1-x))^(3*k+2)/(3*k+2)!))))

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

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(-2^(1/3) * log(1-x))/(2^(2/3)).

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

A357783 a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * Stirling2(n,3*k+1).

Original entry on oeis.org

0, 1, 1, 1, 3, 21, 131, 705, 3515, 17389, 91739, 547889, 3746227, 28241373, 224124083, 1821051233, 15023818091, 126366334125, 1094358852075, 9858890038513, 92983173940419, 918408372280477, 9454438841355395, 100728532687727329, 1103649166937235259

Offset: 0

Seiichi Manyama, Oct 13 2022

nonn

Eric Weisstein's MathWorld, Bell Polynomial.

Cf. A357782, A357784.

Cf. A143816, A357832.

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

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

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)+w^2*Bell_poly(n, v*w)+w*Bell_poly(n, v*w^2))/(3*v));

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). A357782(n) = A(n), a(n) = B(n) and A357784(n) = C(n).
Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(2^(1/3) * (exp(x)-1))/(2^(1/3)).
a(n) = ( Bell_n(2^(1/3)) + w^2 * Bell_n(2^(1/3)*w) + w * Bell_n(2^(1/3)*w^2) )/(3*2^(1/3)), where Bell_n(x) is n-th Bell polynomial.

cp's OEIS Frontend

A357831 a(n) = Sum_{k=0..floor(n/3)} 2^k * |Stirling1(n,3*k)|.

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A357833 a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * |Stirling1(n,3*k+2)|.

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A357783 a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * Stirling2(n,3*k+1).

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