A226515 -id:A226515 - cp's OEIS frontend

A376390 Expansion of e.g.f. (1/x) * Series_Reversion( x*(2 - exp(x))^3 ).

1, 3, 33, 666, 19923, 795438, 39849549, 2405748978, 170114699247, 13796351753670, 1262691211748865, 128760309960844554, 14478116911623185163, 1779761344294187865198, 237465809999666515842261, 34179385495053448088261154, 5279029838285444642785757415, 870905593631158913782753290198

Offset: 0

Seiichi Manyama, Sep 22 2024

nonn

Index entries for reversions of series

Cf. A367135, A376391.

Cf. A226515.

my(N=20, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(2-exp(x))^3)/x))

a(n) = 3*sum(k=0, n, (3*n+k+2)!*stirling(n, k, 2))/(3*n+3)!;

E.g.f. A(x) satisfies A(x) = 1/(2 - exp(x*A(x)))^3.
E.g.f.: B(x)^3, where B(x) is the e.g.f. of A367135.
a(n) = (3/(3*n+3)!) * Sum_{k=0..n} (3*n+k+2)! * Stirling2(n,k).

A375898 E.g.f. satisfies A(x) = 1 / (2 - exp(x * A(x)^(1/3)))^3.

Original entry on oeis.org

1, 3, 21, 234, 3627, 72498, 1780953, 52013118, 1762754655, 68060512458, 2950869169125, 142006584810918, 7513205987292243, 433548334132153698, 27102592662130603857, 1824854382978573444174, 131676307468686605671623, 10137713081262046098901050

Offset: 0

Seiichi Manyama, Sep 01 2024

nonn

Cf. A052894, A375897.

Cf. A226515.

my(N=20, x='x+O('x^N)); Vec(serlaplace((serreverse(x*(2-exp(x)))/x)^3))

a(n) = 3*sum(k=0, n, (n+k+2)!*stirling(n, k, 2))/(n+3)!;

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A052894.
E.g.f.: A(x) = ( (1/x) * Series_Reversion(x * (2 - exp(x))) )^3.
a(n) = (3/(n+3)!) * Sum_{k=0..n} (n+k+2)! * Stirling2(n,k).

A377398 Expansion of e.g.f. (2 - exp(x))^3.

Original entry on oeis.org

1, -3, 3, 9, 3, -63, -357, -1431, -5037, -16623, -52917, -164871, -506877, -1545183, -4684677, -14152311, -42653517, -128353743, -385847637, -1159115751, -3480492957, -10447770303, -31355893797, -94092847191, -282328873197, -847087282863, -2541463175157

Offset: 0

Seiichi Manyama, Oct 27 2024

Seiichi Manyama, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A226515, A377399.

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

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

a(n) = 5*a(n-1) - 6*a(n-2) - 24 for n > 2.
a(n) = Sum_{k=0..3} (-1)^k * k! * binomial(3,k) * Stirling2(n,k).
a(n) = Sum_{k=0..3} (-1)^k * 2^(3-k) * binomial(3,k) * k^n.
G.f.: (1-4*x) * (1-5*x+12*x^2)/((1-x) * (1-2*x) * (1-3*x)).
a(n) = 3*2^(n+1) - 3^n - 12 for n > 0. - Stefano Spezia, Oct 27 2024
a(0) = 1; a(n) = Sum_{k=1..n} (1 - 4 * k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Oct 27 2024

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A376390 Expansion of e.g.f. (1/x) * Series_Reversion( x*(2 - exp(x))^3 ).

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A375898 E.g.f. satisfies A(x) = 1 / (2 - exp(x * A(x)^(1/3)))^3.

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A377398 Expansion of e.g.f. (2 - exp(x))^3.

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