A360479 -id:A360479

Search: a360479 -id:a360479

Displaying 1-3 of 3 results found.

page 1

A360699

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

+10
5

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

LINKS

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

FORMULA

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)).

MATHEMATICA

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}]]

CROSSREFS

Cf. A360592, A360707, A360479.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Feb 16 2023

STATUS

approved

A360747

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

+10
4

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..600

Vaclav Kotesovec, Graph - the asymptotic ratio (30000 terms)

FORMULA

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

MATHEMATICA

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 *)

PROG

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

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

CROSSREFS

Cf. A360479, A360592, A360707.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Feb 19 2023

STATUS

approved

A360748

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

+10
3

1, 1, 1, 2, 5, 10, 21, 53, 133, 327, 861, 2361, 6469, 18168, 52757, 155221, 463077, 1412656, 4379917, 13747504, 43834213, 141866555, 464650309, 1541008295, 5176660997, 17586913779, 60400627453, 209746820056, 735953607173, 2607716976945, 9330605338485

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

LINKS

Table of n, a(n) for n=0..30.

Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

FORMULA

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

a(n) ~ exp(exp(2/3)*n^(2/3)/3^(2/3) - 5*exp(4/3)*n^(1/3)/(18*3^(1/3)) + 22*exp(2)/81) * n^(n/3) / 3^(n/3 + 1) * (1 + (2*exp(2/3)/3^(5/3) - 3295*exp(8/3)/(2916*3^(2/3)))/n^(1/3) + (3^(2/3)/(8*exp(2/3)) + 35*exp(4/3)/(36*3^(1/3)) + 27379*exp(10/3)/(17496*3^(1/3)) + 10857025*exp(16/3)/(51018336*3^(1/3)))/n^(2/3)). - Vaclav Kotesovec, Feb 20 2023

MATHEMATICA

Join[{1}, Table[Sum[Binomial[n - 2*k, k] * (n - 2*k)^k, {k, 0, n/3}], {n, 1, 30}]] (* Vaclav Kotesovec, Feb 20 2023 *)

PROG

(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+k*x^2))^k))

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

CROSSREFS

Cf. A360592, A360749.

Cf. A000930, A360730, A360479.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Feb 19 2023

STATUS

approved

page 1

Search completed in 0.003 seconds