A360699 -id:A360699 - cp's OEIS frontend

A360592 G.f.: Sum_{k>=0} (1 + kx)^k x^k.

1, 1, 2, 5, 14, 44, 149, 543, 2096, 8539, 36444, 162380, 752181, 3612037, 17933038, 91843329, 484280386, 2624400428, 14595111277, 83178971707, 485218783724, 2893881790823, 17628815344600, 109585578277012, 694575012732989, 4485139961090153, 29486515600393930

Offset: 0

Vaclav Kotesovec, Feb 13 2023

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..760

Cf. A360232, A092366, A187018, A360699, A360707.

N:= 40:
S:= series(add((1+k*x)^k*x^k, k=0..N),x,N+1):
seq(coeff(S,x,k),k=0..N); # Robert Israel, Feb 13 2023

nmax = 30; CoefficientList[Series[Sum[(1 + k*x)^k * x^k, {k, 0, nmax}], {x, 0, nmax}], x]
Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^k, {k, 0, n/2}], {n, 1, 30}]}]

{a(n) = polcoeff(sum(m=0, n, (1 + m*x)^m * x^m + x*O(x^n)), n)};
for(n=0, 30, print1(a(n), ", "))

a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k) * (n-k)^k.

a(n) ~ exp(exp(1/2)*sqrt(n/2) - 3*exp(1)/8) * n^(n/2) / 2^(n/2 + 1) * (1 + ((exp(1/2) + exp(-1/2))/2^(5/2) + 11*exp(3/2)/2^(9/2))/sqrt(n)).

A360707 G.f.: Sum_{k>=0} (1 + kx)^k x^(3*k).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 4, 4, 1, 9, 27, 28, 16, 96, 257, 281, 250, 1251, 3161, 3665, 4321, 19489, 47685, 58662, 84099, 354739, 852216, 1110344, 1837924, 7401269, 17604002, 24221890, 44761045, 174287005, 412627144, 597640105, 1204831674, 4574415066, 10818841343

Offset: 0

Vaclav Kotesovec, Feb 17 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A360592, A360699, A360747.

nmax = 50; CoefficientList[Series[Sum[(1 + k*x)^k * x^(3*k), {k, 0, nmax}], {x, 0, nmax}], x]
Join[{1}, Table[Sum[Binomial[k, n - 3*k] * k^(n - 3*k), {k, 0, n}], {n, 1, 50}]]

a(n) = Sum_{k=0..n} binomial(k,n-3*k) * k^(n-3*k).
log(a(n)) ~ n/4 * log(n/4).
a(n) ~ exp(exp(1/4)*n^(1/4)/4^(1/4)) * n^(n/4) / 4^(n/4 + 1) * (1 + 1/(2^(5/2)*exp(1/4)*n^(1/4)) + (67/(192*exp(1/2)) - 15*exp(1/2)/16)/sqrt(n)).

A360479 Expansion of Sum_{k>=0} (x * (1 + (k * x)^2))^k.

Original entry on oeis.org

1, 1, 1, 2, 9, 28, 81, 369, 1753, 7323, 36337, 207401, 1114345, 6308368, 40326033, 256982157, 1658573497, 11650405774, 83966740913, 608348063576, 4659734909385, 36973835868521, 295709600709585, 2454457098977559, 21106884235025305

Offset: 0

Seiichi Manyama, Feb 19 2023

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..635
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A360592, A360747, A360699.

Join[{1}, Table[Sum[Binomial[n - 2*k,k] * (n - 2*k)^(2*k), {k,0,n/3}], {n,1,30}]] (* Vaclav Kotesovec, Feb 19 2023 *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+(k*x)^2))^k))

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

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

a(n) ~ exp(exp(4/3)*n^(1/3)/3^(1/3)) * n^(2*n/3) / 3^(2*n/3 + 1) * (1 + (3^(1/3)/(8*exp(4/3)) - 13*exp(8/3)/(6*3^(2/3))) / n^(1/3) + (67/(128*3^(1/3)*exp(8/3)) - 5*3^(2/3)*exp(4/3)/16 + 169*exp(16/3)/(216*3^(1/3))) / n^(2/3) + (3929/2304 + 497/(1024*exp(4)) + 7913*exp(4)/1728 - 2197*exp(8)/11664)/n). - Vaclav Kotesovec, Feb 19 2023

A360708 Expansion of Sum_{k>=0} (x^2 / (1 - k*x))^k.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 14, 42, 136, 479, 1825, 7433, 32053, 145608, 695081, 3479117, 18209842, 99373513, 563920590, 3320674902, 20255823092, 127799984935, 832807892861, 5597481205009, 38753768384761, 276057156622776, 2021100095469577, 15193591060371577

Offset: 0

Seiichi Manyama, Feb 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..630

Cf. A080108, A360709.

Cf. A000248, A360699.

Join[{1},Table[Sum[Binomial[n-k-1,k-1] * k^(n-2*k), {k,0,n/2}], {n,1,40}]] (* Vaclav Kotesovec, Feb 20 2023 *)

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

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

a(n) = Sum_{k=1..floor(n/2)} k^(n-2*k) * binomial(n-k-1,k-1) for n > 0.

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A360592 G.f.: Sum_{k>=0} (1 + k*x)^k * x^k.

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A360707 G.f.: Sum_{k>=0} (1 + k*x)^k * x^(3*k).

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A360479 Expansion of Sum_{k>=0} (x * (1 + (k * x)^2))^k.

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A360708 Expansion of Sum_{k>=0} (x^2 / (1 - k*x))^k.

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A360592 G.f.: Sum_{k>=0} (1 + kx)^k x^k.

A360707 G.f.: Sum_{k>=0} (1 + kx)^k x^(3*k).