A377204 -id:A377204 - cp's OEIS frontend

A377213 Expansion of 1/(1 - 4*x^3/(1-x))^(3/2).

1, 0, 0, 6, 6, 6, 36, 66, 96, 266, 576, 1026, 2246, 4866, 9516, 19598, 41286, 83526, 170048, 351378, 716850, 1458098, 2984028, 6087270, 12380900, 25224222, 51356400, 104380510, 212164362, 431148222, 875353220, 1776567762, 3604752672, 7310374010, 14819370480, 30033014994

Offset: 0

Seiichi Manyama, Oct 20 2024

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1800

Cf. A377197, A377204.

Cf. A360309, A377216.

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^3/(1-x))^(3/2))); // Vincenzo Librandi, May 08 2025

Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-2*k-1,n-3*k],{k,0,Floor[n/3]}],{n,0,35}] (* Vincenzo Librandi, May 08 2025 *)

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

a(n) = (2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n+3)*a(n-3) - 2*(2*n-2)*a(n-4))/n for n > 3.
a(n) = Sum_{k=0..floor(n/2)} (2*k+1) * binomial(2*k,k) * binomial(n-2*k-1,n-3*k).
a(n) ~ sqrt(n) * 2^(n-2) / sqrt(Pi). - Vaclav Kotesovec, May 03 2025

A377215 Expansion of 1/(1 - 4*x^2/(1-x))^(5/2).

Original entry on oeis.org

1, 0, 10, 10, 80, 150, 640, 1550, 5190, 13870, 41912, 115650, 333490, 925970, 2607540, 7220062, 20053700, 55230870, 152005380, 416295350, 1137980678, 3100453710, 8429823180, 22862244210, 61882724100, 167159512794, 450739897980, 1213298505770, 3260824389510

Offset: 0

Seiichi Manyama, Oct 20 2024

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..1500

Cf. A377199, A377216.

Cf. A026585, A377204.

R:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^2/(1-x))^(5/2))); // Vincenzo Librandi, May 08 2025

Table[Sum[(-4)^k*Binomial[-5/2,k]*Binomial[n-k-1,n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* Vincenzo Librandi, May 08 2025 *)

a(n) = sum(k=0, n\2, (-4)^k*binomial(-5/2, k)*binomial(n-k-1, n-2*k));

a(n) = (2*(n-1)*a(n-1) + (3*n+14)*a(n-2) - 2*(2*n-1)*a(n-3))/n for n > 2.
a(n) = Sum_{k=0..floor(n/2)} (-4)^k * binomial(-5/2,k) * binomial(n-k-1,n-2*k).
a(n) ~ n^(3/2) * 2^(3*n - 1/2) / (3 * 17^(5/4) * sqrt(Pi) * (sqrt(17) - 1)^(n - 5/2)). - Vaclav Kotesovec, May 03 2025

A383499 Expansion of 1/sqrt( (1-x) * (1-x-4*x^2)^3 ).

Original entry on oeis.org

1, 2, 9, 22, 71, 186, 537, 1434, 3957, 10586, 28603, 76266, 203767, 540986, 1435533, 3796050, 10026015, 26422350, 69544765, 182759750, 479731113, 1257750486, 3294264627, 8619879726, 22535782953, 58869786162, 153671378139, 400861115498, 1045005290059, 2722601576322

Offset: 0

Seiichi Manyama, May 05 2025

nonn

Cf. A383254, A383503.

Cf. A026569, A377204.

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

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

a(n) ~ sqrt(n/Pi) * ((1 + sqrt(17))/2)^(n + 5/2) / 17^(3/4). - Vaclav Kotesovec, May 05 2025

cp's OEIS Frontend

A377213 Expansion of 1/(1 - 4*x^3/(1-x))^(3/2).

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A377215 Expansion of 1/(1 - 4*x^2/(1-x))^(5/2).

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A383499 Expansion of 1/sqrt( (1-x) * (1-x-4*x^2)^3 ).

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