A360707 -id:A360707 - 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)).

A360699 G.f.: Sum_{k>=0} (1 + kx)^k x^(2*k).

Original entry on oeis.org

1, 0, 1, 1, 1, 4, 5, 9, 28, 43, 97, 281, 507, 1286, 3666, 7494, 20470, 58725, 132484, 381700, 1113180, 2719887, 8171219, 24337511, 63524916, 197606643, 602261524, 1662206380, 5328738685, 16628469912, 48148703533, 158544768073, 506473892417, 1529218062752, 5159071807165

Offset: 0

Vaclav Kotesovec, Feb 16 2023

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A360592, A360707, A360479.

nmax = 40; CoefficientList[Series[Sum[(1 + k*x)^k * x^(2*k), {k, 0, nmax}], {x, 0, nmax}], x]
Join[{1}, Table[Sum[Binomial[k, n - 2*k] * k^(n - 2*k), {k, 0, n}], {n, 1, 40}]]

a(n) = Sum_{k=0..n} binomial(k,n-2*k) * k^(n-2*k).
log(a(n)) ~ n/3 * log(n/3).
a(n) ~ exp(exp(1/3)*n^(1/3)/3^(1/3)) * n^(n/3) / 3^(n/3 + 1) * (1 + (3^(1/3)/(8*exp(1/3)) - 4*exp(2/3)/3^(5/3)) / n^(1/3) + (67/(128*3^(1/3)*exp(2/3)) + 8*exp(4/3)/3^(10/3)) / n^(2/3)).

A360709 Expansion of Sum_{k>=0} (x^3 / (1 - k*x))^k.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 5, 13, 34, 90, 247, 720, 2256, 7568, 26814, 98982, 377541, 1484254, 6021789, 25271173, 109850447, 494355359, 2298362532, 11008133629, 54175202125, 273460921605, 1414449612648, 7494262602464, 40669492399396

Offset: 0

Seiichi Manyama, Feb 17 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..692

Cf. A080108, A360708.

Cf. A078012, A360707.

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

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

a(n) = Sum_{k=1..floor(n/3)} k^(n-3*k) * binomial(n-2*k-1,k-1) for n > 0.

A360747 Expansion of Sum_{k>=0} (x * (1 + (k * x)^3))^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 17, 82, 257, 690, 3484, 26978, 160347, 726085, 3529206, 26885924, 220706533, 1474182023, 8834370165, 65392181686, 604821608674, 5230627589958, 39543579302104, 312733691925723, 3013530105191283, 30474809255061289

Offset: 0

Seiichi Manyama, Feb 19 2023

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..600
Vaclav Kotesovec, Graph - the asymptotic ratio (30000 terms)

Cf. A360479, A360592, A360707.

Join[{1},Table[Sum[Binomial[n - 3*k,k] * (n - 3*k)^(3*k), {k,0,n/4}], {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)^3))^k))

a(n) = sum(k=0, n\4, (n-3*k)^(3*k)*binomial(n-3*k, k));

a(n) = Sum_{k=0..floor(n/4)} (n-3*k)^(3*k) * binomial(n-3*k,k).

a(n) ~ exp(exp(9/4)*n^(1/4)/sqrt(2)) * n^(3*n/4) / 2^(3*n/2 + 2) * (1 + 1/(4*sqrt(2)*exp(9/4) * n^(1/4)) + (67/(192*exp(9/2)) - 37*exp(9/2)/16) / sqrt(n) + (497/(768*sqrt(2)*exp(27/4)) - 205*exp(9/4)/(64*sqrt(2))) / n^(3/4) + (10721/3072 + 218831/(368640*exp(9)) + (1369*exp(9))/512)/n), see graph for more minor asymptotic terms. - Vaclav Kotesovec, Feb 20 2023

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

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A360699 G.f.: Sum_{k>=0} (1 + k*x)^k * x^(2*k).

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A360709 Expansion of Sum_{k>=0} (x^3 / (1 - k*x))^k.

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A360747 Expansion of Sum_{k>=0} (x * (1 + (k * x)^3))^k.

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

A360699 G.f.: Sum_{k>=0} (1 + kx)^k x^(2*k).