A360151 -id:A360151

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A105872

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

+10
11

1, 2, 6, 21, 75, 273, 1009, 3770, 14202, 53846, 205216, 785460, 3017106, 11624580, 44905518, 173863965, 674506059, 2621371005, 10203609597, 39773263035, 155231706951, 606554343495, 2372544034143, 9289131196485, 36401388236461

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

OFFSET

0,2

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..1000

FORMULA

G.f.: 2/(4*x^2+sqrt(1-4*x)*(3*x+1)-5*x+1). - Vladimir Kruchinin, May 24 2014

Conjecture: -3*(n+1)*(7*n-2)*a(n) +6*(7*n+5)*(2*n-1)*a(n-1) -(n+1)*(7*n-2)*a(n-2) +2*(7*n+5)*(2*n-1)*a(n-3)=0. - R. J. Mathar, Nov 28 2014

a(n) ~ 2^(2*n+3) / (7*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 28 2023

a(n) = [x^n] 1/((1-x^3) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024

MATHEMATICA

Table[Sum[Binomial[2n-3k, n], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Harvey P. Dale, Jan 13 2015 *)

PROG

(PARI) a(n) = sum(k=0, n\3, binomial(2*n-3*k, n)); \\ Seiichi Manyama, Jan 28 2023

CROSSREFS

Cf. A144904, A360150, A360151, A360152, A360153.

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Apr 23 2005

EXTENSIONS

Erroneous title changed by Paul Barry, Apr 14 2010

Name corrected by Seiichi Manyama, Jan 28 2023

STATUS

approved

A360150

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

+10
11

1, 2, 6, 21, 77, 288, 1090, 4159, 15964, 61557, 238221, 924597, 3597290, 14024341, 54770176, 214218966, 838959762, 3289471537, 12910910288, 50720828034, 199422778415, 784672001097, 3089564308849, 12172411084432, 47984843655991, 189260578353602

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..25.

FORMULA

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^3 * c(x)^5) ), where c(x) is the g.f. of A000108.

a(n) ~ 2^(2*n+1) / sqrt(Pi*n). - Vaclav Kotesovec, Jan 28 2023

D-finite with recurrence n*a(n) +2*(-7*n+6)*a(n-1) +2*(36*n-61)*a(n-2) +4*(-41*n+103)*a(n-3) +(161*n-530)*a(n-4) +(-71*n+278)*a(n-5) +6*(2*n-9)*a(n-6)=0. - R. J. Mathar, Mar 12 2023

a(n) = [x^n] 1/(((1-x)^2-x^3) * (1-x)^(n-1)). - Seiichi Manyama, Apr 09 2024

MAPLE

A360150 := proc(n)

add(binomial(2*n-k, n-3*k), k=0..n/3) ;

end proc:

seq(A360150(n), n=0..70) ; # R. J. Mathar, Mar 12 2023

MATHEMATICA

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

PROG

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

(PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1-x^3*(2/(1+sqrt(1-4*x)))^5)))

CROSSREFS

Cf. A105872, A144904, A360151, A360152, A360153, A360168.

Cf. A000108.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jan 28 2023

STATUS

approved

A360152

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

+10
7

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

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..25.

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

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

CROSSREFS

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

Cf. A000108.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jan 28 2023

STATUS

approved

A360153

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

+10
7

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

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..25.

FORMULA

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

MAPLE

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

MATHEMATICA

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 *)

PROG

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

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

CROSSREFS

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

Cf. A006134, A106188.

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Jan 28 2023

STATUS

approved

A360168

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

+10
4

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

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

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..25.

FORMULA

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

MAPLE

A360168 := proc(n)

add(binomial(2*n, n-3*k), k=0..n/3) ;