A360592 -id:A360592 - cp's OEIS frontend

A360611 Expansion of Sum_{k>=0} (k * x * (1 + x))^k.

1, 1, 5, 35, 341, 4230, 63844, 1135753, 23273363, 539881365, 13986073419, 400227436252, 12538263892232, 426810214125441, 15687071552060221, 619144491880324087, 26117514728711229877, 1172635546310430028562, 55833864788507320490268

Offset: 0

Seiichi Manyama, Feb 14 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..385

Cf. A355494, A360592.

Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^(n-k), {k, 0, n/2}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 14 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0,N, (k*x*(1+x))^k))

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

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

a(n) ~ exp(exp(-1)) * n^n. - Vaclav Kotesovec, Feb 14 2023

A360618 Expansion of Sum_{k>=0} (k * x * (1 + k*x))^k.

Original entry on oeis.org

1, 1, 5, 43, 515, 7950, 150086, 3349945, 86296849, 2519907605, 82249222661, 2967449372028, 117266100841668, 5037282382077353, 233701540415817409, 11645959855678136519, 620389246928233860127, 35181554115178393462386, 2116059351692554708911298

Offset: 0

Seiichi Manyama, Feb 14 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..373

Cf. A072034, A360592, A360611.

Flatten[{1, Table[Sum[Binomial[n-k, k] * (n-k)^n, {k, 0, n/2}], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 14 2023 *)

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x*(1+k*x))^k))

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

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

a(n) ~ c * d^n * n^n, where d = (1-r)^(2-r) / (r^r * (1-2*r)^(1-2*r)) where r = 0.163662210494891118101893756356803907477984542... is the root of the equation (1-2*r)^2 = r*(1-r) * exp(1/(1-r)) and c = 0.78619174295244329885973980954744130517052330684023764340463604028671858569... - Vaclav Kotesovec, Feb 14 2023

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)).

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)).

A360782 Expansion of Sum_{k>=0} x^k / (1 - k*x^2)^(k+1).

Original entry on oeis.org

1, 1, 1, 3, 7, 16, 45, 125, 363, 1127, 3561, 11696, 39727, 138113, 494213, 1811075, 6784115, 25985928, 101520833, 404305549, 1640002039, 6767576175, 28395916893, 121048681024, 523902418555, 2300906314849, 10248029334297, 46266088140291

Offset: 0

Seiichi Manyama, Feb 20 2023

Cf. A000248, A360783.

Cf. A000045, A360592.

Join[{1}, Table[Sum[Binomial[n-k,k] * (n-2*k)^k, {k,0,n/2}], {n,1,30}]] (* Vaclav Kotesovec, Feb 21 2023 *)

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

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

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

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

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

A360787 Expansion of Sum_{k>=0} x^k / (1 - (k*x)^2)^(k+1).

Original entry on oeis.org

1, 1, 1, 3, 13, 40, 177, 965, 4733, 28103, 184065, 1191888, 8713549, 67005689, 528870257, 4526024267, 40051790333, 368513578472, 3583302492545, 35868588067501, 373781214260749, 4052932682659599, 45218033687522481, 523234757502985824, 6245693941097387773

Offset: 0

Seiichi Manyama, Feb 20 2023

Cf. A000248, A360788.

Cf. A360782, A360592.

Join[{1}, Table[Sum[Binomial[n-k,k] * (n-2*k)^(2*k), {k,0,n/2}], {n,1,30}]] (* Vaclav Kotesovec, Feb 21 2023 *)

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

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

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

A360748 Expansion of Sum_{k>=0} (x * (1 + k*x^2))^k.

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Feb 19 2023

nonn

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

Cf. A360592, A360749.

Cf. A000930, A360730, A360479.

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 *)

my(N=40, 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)^k*binomial(n-2*k, k));

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

A360232 G.f. Sum_{n>=0} a(n)x^n = Sum_{n>=0} (1 + nx + x^2)^n * x^n.

Original entry on oeis.org

1, 1, 2, 6, 16, 51, 172, 626, 2409, 9791, 41671, 185224, 855865, 4100761, 20314349, 103827684, 546388333, 2955518901, 16407286272, 93350267922, 543674327227, 3237568471183, 19693508812475, 122249256779882, 773797772369256, 4990290667614087, 32766888950422831

Offset: 0

Paul D. Hanna, Feb 12 2023

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 16*x^4 + 51*x^5 + 172*x^6 + 626*x^7 + 2409*x^8 + 9791*x^9 + 41671*x^10 + 185224*x^11 + 855865*x^12 + ...
where
A(x) = 1 + (1 + x + x^2)*x + (1 + 2*x + x^2)^2*x^2 + (1 + 3*x + x^2)^3*x^3 + (1 + 4*x + x^2)^4*x^4 + ... + (1 + n*x + x^2)^n*x^n + ...

Paul D. Hanna, Table of n, a(n) for n = 0..500

Cf. A294573, A360348, A360592.

nmax = 30; CoefficientList[Series[Sum[(1 + k*x + x^2)^k * x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 13 2023 *)
Flatten[{1, Table[Sum[Sum[Binomial[k,j] * Binomial[j,n-k-j] * k^(2*j - n + k), {j, 0, k}], {k, 1, n}], {n, 1, 30}]}] (* Vaclav Kotesovec, Feb 14 2023 *)

{a(n) = polcoeff( sum(m=0,n, (1 + m*x + x^2)^m * x^m +x*O(x^n)),n)}
for(n=0,30,print1(a(n),", "))

a(n) = Sum_{k=1..n}(Sum_{j=0..k} binomial(k,j) * binomial(j,n-k-j) * k^(2*j-n+k)). - Vaclav Kotesovec, Feb 14 2023

cp's OEIS Frontend

A360611 Expansion of Sum_{k>=0} (k * x * (1 + x))^k.

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A360618 Expansion of Sum_{k>=0} (k * x * (1 + k*x))^k.

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

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

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A360782 Expansion of Sum_{k>=0} x^k / (1 - k*x^2)^(k+1).

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

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

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A360787 Expansion of Sum_{k>=0} x^k / (1 - (k*x)^2)^(k+1).

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

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

A360232 G.f. Sum_{n>=0} a(n)x^n = Sum_{n>=0} (1 + nx + x^2)^n * x^n.