A382514 -id:A382514 - cp's OEIS frontend

A382536 Expansion of 1/(1 - x(1 + 4x)^(3/2)).

1, 1, 7, 19, 63, 221, 679, 2365, 7499, 25351, 82043, 274031, 892263, 2972127, 9686899, 32261819, 105124711, 350277365, 1140610399, 3803874525, 12372800403, 41319077557, 134176480535, 448958154449, 1454582791283, 4879992151217, 15762304059447, 53067612190093

Offset: 0

Seiichi Manyama, Mar 31 2025

a(62) is negative.

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Cf. A362154, A382537, A382538.

Cf. A002421, A382514.

R := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + 4*x)^(3/2)); seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // Vincenzo Librandi, Apr 01 2025

Table[Sum[4^(n-k)*Binomial[3*k/2,n-k],{k,0,n}],{n,0,35}] (* Vincenzo Librandi, Apr 01 2025 *)

a(n) = sum(k=0, n, 4^(n-k)*binomial(3*k/2, n-k));

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(3*k/2,n-k).
D-finite with recurrence (-n+1)*a(n) +2*(-2*n+7)*a(n-1) +(n-1)*a(n-2) +2*(8*n-13)*a(n-3) +24*(4*n-9)*a(n-4) +32*(8*n-23)*a(n-5) +128*(2*n-7)*a(n-6)=0. - R. J. Mathar, Apr 02 2025
a(n) ~ 3 * (-1)^(n+1) * 2^(2*n-4) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Apr 13 2025

A382515 Expansion of 1/(1 - x/(1 - 4*x)^(5/2)).

Original entry on oeis.org

1, 1, 11, 91, 691, 5101, 37323, 272405, 1987047, 14493479, 105718071, 771148119, 5625136651, 41032826127, 299316769887, 2183389173811, 15926906427179, 116180104751925, 847485191674867, 6182049517420133, 45095462188117951, 328952511222499589, 2399570809473795931

Offset: 0

Seiichi Manyama, Mar 30 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A026671, A382514.

Cf. A002802.

Table[Sum[4^(n-k)*Binomial[n+3*k/2-1,n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Mar 30 2025 *)

a(n) = sum(k=0, n, 4^(n-k)*binomial(n+3*k/2-1, n-k));

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n+3*k/2-1,n-k).

D-finite with recurrence (-n+1)*a(n) +2*(12*n-19)*a(n-1) +(-239*n+519)*a(n-2) +2*(638*n-1751)*a(n-3) +1280*(-3*n+10)*a(n-4) +512*(12*n-47)*a(n-5) +2048*(-2*n+9)*a(n-6)=0. - R. J. Mathar, Mar 31 2025

A382542 Expansion of 1/(1 - x/(1 - 4*x)^(3/2))^3.

Original entry on oeis.org

1, 3, 24, 172, 1191, 8091, 54214, 359274, 2358945, 15365815, 99399132, 639081780, 4086689187, 26006041209, 164767882902, 1039787209898, 6537976304109, 40973438195025, 255998969164612, 1594973077037136, 9911483124031335, 61443351455986359, 380044418794190118

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Cf. A382514, A382541.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 4*x)^(3/2))^3)); // Vincenzo Librandi, May 12 2025

Table[Sum[(4)^(n-k)* Binomial[k+2,2]*Binomial[n+k/2-1, n-k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, May 12 2025 *)

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

a(n) = Sum_{k=0..n} 4^(n-k) * binomial(k+2,2) * binomial(n+k/2-1,n-k).

D-finite with recurrence (1951*n-648)*(n-1)*a(n) +2*(-19507*n^2+55239*n-42212)*a(n-1) +(310113*n^2-1357025*n+1557754)*a(n-2) +2*(-616231*n^2+3607803*n-5547102)*a(n-3) +4*(616138*n^2-4491203*n+8306122)*a(n-4) -256*(3899*n-16903)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Apr 02 2025

A382541 Expansion of 1/(1 - x/(1 - 4*x)^(3/2))^2.

Original entry on oeis.org

1, 2, 15, 100, 645, 4098, 25795, 161256, 1002513, 6203434, 38230951, 234774948, 1437193101, 8773022374, 53416562787, 324488659784, 1967025910873, 11901070329414, 71878009609591, 433411746865948, 2609477469570885, 15689257525890666, 94208451895149123

Offset: 0

Seiichi Manyama, Mar 31 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Cf. A382514, A382542.

Cf. A382539.

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1-4*x)^(3/2))^2)); // Vincenzo Librandi, May 12 2025

Table[Sum[(4)^(n-k)* (k+1)* Binomial[n+k/2-1,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, May 12 2025 *)

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

a(n) = Sum_{k=0..n} 4^(n-k) * (k+1) * binomial(n+k/2-1,n-k).

D-finite with recurrence (-n+1)*a(n) +2*(8*n-13)*a(n-1) +(-95*n+217)*a(n-2) +2*(126*n-359)*a(n-3) +128*(-2*n+7)*a(n-4)=0. - R. J. Mathar, Apr 02 2025

cp's OEIS Frontend

A382536 Expansion of 1/(1 - x*(1 + 4*x)^(3/2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A382515 Expansion of 1/(1 - x/(1 - 4*x)^(5/2)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A382542 Expansion of 1/(1 - x/(1 - 4*x)^(3/2))^3.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A382541 Expansion of 1/(1 - x/(1 - 4*x)^(3/2))^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A382536 Expansion of 1/(1 - x(1 + 4x)^(3/2)).