A144904 -id:A144904 - cp's OEIS frontend

A226405 Expansion of x/((1-x-x^3)*(1-x)^3).

0, 1, 4, 10, 21, 40, 71, 120, 196, 312, 487, 749, 1139, 1717, 2571, 3830, 5683, 8407, 12408, 18281, 26898, 39537, 58071, 85245, 125082, 183478, 269074, 394534, 578418, 847927, 1242926, 1821840, 2670295, 3913782, 5736217, 8407142, 12321590, 18058510, 26466393

Offset: 0

Alois P. Heinz, Jun 06 2013

From Bruno Berselli, Jun 07 2013: (Start)
A050228(n)  = a(n)   -a(n-1), n>0.
A077868(n-1)= a(n) -2*a(n-1)   +a(n-2), n>1.
A000217(n)  = a(n)   -a(n-1)             -a(n-3), n>2.
A000930(n-1)= a(n) -3*a(n-1) +3*a(n-2)   -a(n-3), n>2.
n           = a(n) -2*a(n-1)   +a(n-2)   -a(n-3)   +a(n-4), n>3.
1           = a(n) -3*a(n-1) +3*a(n-2) -2*a(n-3) +2*a(n-4)   -a(n-5), n>4.
0           = a(n) -4*a(n-1) +6*a(n-2) -5*a(n-3) +4*a(n-4) -3*a(n-5) +a(n-6), n>5.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-4,3,-1).

4th column of A144903.
Cf. A000217, A078012, A099567, A135851.
Cf. A000930, A050228, A077868, A144898, A144899, A144900, A144901, A144902, A144903, A144904, A226405.

A226405:= func< n | n eq 0 select 0 else (&+[Binomial(n-2*j+2, j+3): j in [0..Floor((n+2)/3)]]) >;
[A226405(n): n in [0..40]]; // G. C. Greubel, Jul 27 2022

a:= n-> (Matrix(6, (i, j)-> if i=j-1 then 1 elif j=1 then [4, -6, 5, -4, 3, -1][i] else 0 fi)^n)[1, 2]: seq(a(n), n=0..40);

LinearRecurrence[{4,-6,5,-4,3,-1}, {0,1,4,10,21,40}, 40]  (* Bruno Berselli, Jun 07 2013 *)
CoefficientList[Series[x/((1-x-x^3)*(1-x)^3), {x, 0, 50}], x] (* G. C. Greubel, Apr 28 2017 *)

my(x='x+O('x^50)); Vec(x/((1-x-x^3)*(1-x)^3)) \\ G. C. Greubel, Apr 28 2017

def A226405(n): return sum(binomial(n-2*j+2, j+3) for j in (0..((n+2)//3)))
[A226405(n) for n in (0..40)] # G. C. Greubel, Jul 27 2022

G.f.: x/((1-x-x^3)*(1-x)^3).
From G. C. Greubel, Jul 27 2022: (Start)
a(n) = Sum_{j=0..floor((n+2)/3)} binomial(n-2*j+2, j+3).
a(n) = A099567(n+2, 3). (End)

A360152 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-5k,n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 73, 262, 960, 3562, 13347, 50393, 191406, 730555, 2799622, 10765092, 41513751, 160490906, 621805286, 2413738744, 9385635299, 36550685683, 142534105563, 556514122937, 2175296066129, 8511430278018, 33334299581686, 130662787246407

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A105872, A144904, A360150, A360151, A360153.

Cf. A000108.

A360152 := proc(n)
    add(binomial(2*n-5*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360152(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a[n_] := Sum[Binomial[2*n - 5*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)

a(n) = sum(k=0, n\3, binomial(2*n-5*k, n-3*k));

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-2*x^3/(1+sqrt(1-4*x)))))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)) ), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+5) / (31 * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence 2*n*a(n) +4*(-2*n+1)*a(n-1) +(-3*n+4)*a(n-2) +2*(6*n-11)*a(n-3) +(n-4)*a(n-4) +2*(-n+9)*a(n-5) +4*(-2*n+1)*a(n-6) +(-n+4)*a(n-7) +2*(2*n-9)*a(n-8)=0. - R. J. Mathar, Mar 12 2023

A360153 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-6k,n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 72, 258, 945, 3504, 13128, 49565, 188260, 718560, 2753721, 10588860, 40835160, 157871241, 611669250, 2374441380, 9233006541, 35956933050, 140220970200, 547490880981, 2140055896770, 8373651697800, 32795094564081, 128550662334522

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A105872, A144904, A360150, A360151, A360152, A360168.

Cf. A006134, A106188.

A360153 := proc(n)
    add(binomial(2*n-6*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360153(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a[n_] := Sum[Binomial[2*n - 6*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)

a(n) = sum(k=0, n\3, binomial(2*n-6*k, n-3*k));

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3) ).
a(n) ~ 2^(2*n + 6) / (63 * sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
a(n)-a(n-3) = A000984(n). - R. J. Mathar, Mar 12 2023
D-finite with recurrence n*a(n) +2*(-2*n+1)*a(n-1) -n*a(n-3) +2*(2*n-1)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A360151 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-4k,n-3*k).

Original entry on oeis.org

1, 2, 6, 21, 74, 267, 981, 3648, 13690, 51744, 196699, 751237, 2880345, 11080081, 42743148, 165291569, 640563158, 2487083484, 9672626600, 37674470433, 146937686295, 573781535775, 2243050091905, 8777451670102, 34379401083017, 134770951530840

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A105872, A144904, A360150, A360152, A360153.

Cf. A000108.

A360151 := proc(n)
    add(binomial(2*n-4*k,n-3*k),k=0..n/3) ;
end proc:
seq(A360151(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a[n_] := Sum[Binomial[2*n - 4*k, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)

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

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^2)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^2) ), where c(x) is the g.f. of A000108.
a(n) ~ 2^(2*n+4) / (15*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence +2*n*a(n) +(-11*n+6)*a(n-1) +(19*n-24)*a(n-2) +2*(-16*n+33)*a(n-3) +2*(11*n-36)*a(n-4) +(-25*n+78)*a(n-5) +6*(n-3)*a(n-6) +4*(-2*n+9)*a(n-7)=0. - R. J. Mathar, Mar 12 2023

A371758 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-3k-1,n-3*k).

Original entry on oeis.org

1, 1, 3, 11, 39, 141, 519, 1933, 7263, 27479, 104543, 399543, 1532779, 5899167, 22766607, 88073091, 341425551, 1326019653, 5158412943, 20096457549, 78396460299, 306190920837, 1197181197567, 4685523856881, 18354865147011, 71962695111841, 282357198103815

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A371770, A371771, A371772.

Cf. A144904, A371773, A371777.

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

a(n) = [x^n] 1/((1-x^3) * (1-x)^n).
a(n) = binomial(2*n-1, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-2*n)/3, 2*(1-n)/3, 1-2*n/3], 1). - Stefano Spezia, Apr 06 2024
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: 3*n*(7*n-11)*a(n) = 6*(2*n-3)*(7*n-4)*a(n-1) - n*(7*n-11)*a(n-2) + 2*(2*n-3)*(7*n-4)*a(n-3).
a(n) ~ 2^(2*n+2) / (7*sqrt(Pi*n)). (End)

A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2n,n-3k).

Original entry on oeis.org

1, 2, 6, 21, 78, 297, 1145, 4447, 17358, 68001, 267141, 1051767, 4148281, 16385111, 64797543, 256515731, 1016368078, 4030114641, 15990813773, 63485616391, 252175202373, 1002136689071, 3984080489263, 15844839393411, 63036297959993, 250855287692647

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A105872, A144904, A360150, A360151, A360152, A360153.
Cf. A000108, A032443, A114121.
Cf. A002450, A371777.

A360168 := proc(n)
    add(binomial(2*n,n-3*k),k=0..n/3) ;
end proc:
seq(A360168(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

a[n_] := Sum[Binomial[2*n, n - 3*k], {k, 0, Floor[n/3]}]; Array[a, 26, 0] (* Amiram Eldar, Jan 28 2023 *)

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

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^6)))

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^6) ), where c(x) is the g.f. of A000108.
D-finite with recurrence n*a(n) +2*(-4*n+3)*a(n-1) +8*(2*n-3)*a(n-2) +3*(-n+2)=0. - R. J. Mathar, Mar 12 2023
a(n) = [x^n] 1/(((1-x)^3-x^3) * (1-x)^(n-2)). - Seiichi Manyama, Apr 10 2024

A371754 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-2k,n-3*k).

Original entry on oeis.org

1, 3, 15, 85, 505, 3081, 19125, 120173, 761995, 4865697, 31244029, 201544551, 1305039209, 8477521051, 55221311565, 360559717807, 2359123470971, 15463951609491, 101530816122729, 667587477393509, 4395294402200983, 28972295880583861, 191181607835416543

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A144904, A371755, A371756.

Cf. A066380, A371742.

Table[Sum[Binomial[3n-2k,n-3k],{k,0,Floor[n/3]}],{n,0,30}] (* Harvey P. Dale, Oct 19 2024 *)

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

a(n) = [x^n] 1/((1-x-x^3) * (1-x)^(2*n)).

a(n) ~ 3^(3*n + 5/2) / (17 * sqrt(Pi*n) * 2^(2*n)). - Vaclav Kotesovec, Apr 05 2024

A371756 a(n) = Sum_{k=0..floor(n/3)} binomial(5n-2k,n-3*k).

Original entry on oeis.org

1, 5, 45, 456, 4863, 53383, 597052, 6765471, 77407257, 892270250, 10346070471, 120542238796, 1410040212166, 16549315766244, 194792566133507, 2298472850258746, 27179673132135409, 322013956853586970, 3821532498419234994, 45420775578132979989

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A144904, A371754, A371755.

a(n) = sum(k=0, n\3, binomial(5*n-2*k, n-3*k));

a(n) = [x^n] 1/((1-x-x^3) * (1-x)^(4*n)).
a(n) ~ 5^(5*n + 5/2) / (99 * sqrt(Pi*n) * 2^(8*n - 1/2)). - Vaclav Kotesovec, Apr 05 2024
a(n) = binomial(5*n, n)*hypergeom([1, (1-n)/3, (2-n)/3, -n/3], [(1-5*n)/2, -5*n/2, 1+4*n], -27/4). - Stefano Spezia, Apr 06 2024

A371872 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-2k-1,n-3*k).

Original entry on oeis.org

1, 1, 3, 11, 40, 147, 547, 2055, 7777, 29602, 113204, 434591, 1673821, 6464539, 25026534, 97087873, 377329971, 1468856383, 5726159811, 22351657810, 87350137071, 341726039806, 1338173763288, 5244830032639, 20573285744475, 80761011408961, 317249771957040

Offset: 0

Seiichi Manyama, Apr 10 2024

nonn

Cf. A360150, A371871, A371873.

Cf. A144904, A371842.

A371872 := proc(n)
    add(binomial(2*n-2*k-1,n-3*k),k=0..floor(n/3)) ;
end proc:
seq(A371872(n),n=0..60) ; # R. J. Mathar, Apr 22 2024

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

a(n) = [x^n] 1/((1-x-x^3) * (1-x)^(n-1)).

D-finite with recurrence +n*a(n) +(-15*n+14)*a(n-1) +3*(27*n-50)*a(n-2) +2*(-93*n+259)*a(n-3) +24*(7*n-26)*a(n-4) +(-69*n+260)*a(n-5) +10*(2*n-9)*a(n-6)=0. - R. J. Mathar, Apr 22 2024

A371755 a(n) = Sum_{k=0..floor(n/3)} binomial(4n-2k,n-3*k).

Original entry on oeis.org

1, 4, 28, 221, 1834, 15657, 136137, 1199014, 10661184, 95493145, 860339723, 7788028028, 70777321331, 645359630071, 5901209474518, 54093485799726, 496910913391428, 4573312196055502, 42160889572810597, 389258294230352460, 3598732127428879981

Offset: 0

Seiichi Manyama, Apr 05 2024

nonn

Cf. A144904, A371754, A371756.

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

a(n) = [x^n] 1/((1-x-x^3) * (1-x)^(3*n)).

a(n) ~ 2^(8*n + 9/2) / (47 * sqrt(Pi*n) * 3^(3*n - 1/2)). - Vaclav Kotesovec, Apr 05 2024

cp's OEIS Frontend

A226405 Expansion of x/((1-x-x^3)*(1-x)^3).

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A360152 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-5*k,n-3*k).

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A360153 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-6*k,n-3*k).

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A360151 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-4*k,n-3*k).

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A371758 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k-1,n-3*k).

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A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,n-3*k).

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A371754 a(n) = Sum_{k=0..floor(n/3)} binomial(3*n-2*k,n-3*k).

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A371756 a(n) = Sum_{k=0..floor(n/3)} binomial(5*n-2*k,n-3*k).

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A360152 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-5k,n-3*k).

A360153 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-6k,n-3*k).

A360151 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-4k,n-3*k).

A371758 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-3k-1,n-3*k).

A360168 a(n) = Sum_{k=0..floor(n/3)} binomial(2n,n-3k).

A371754 a(n) = Sum_{k=0..floor(n/3)} binomial(3n-2k,n-3*k).

A371756 a(n) = Sum_{k=0..floor(n/3)} binomial(5n-2k,n-3*k).

A371872 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-2k-1,n-3*k).

A371755 a(n) = Sum_{k=0..floor(n/3)} binomial(4n-2k,n-3*k).