A360150 -id:A360150 - cp's OEIS frontend

A105872 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-3k, n).

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

Offset: 0

Paul Barry, Apr 23 2005

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

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

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

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

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

Erroneous title changed by Paul Barry, Apr 14 2010

Name corrected by Seiichi Manyama, Jan 28 2023

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

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

A371871 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-3k-2,n-3*k).

Original entry on oeis.org

1, 0, 1, 5, 18, 66, 246, 924, 3493, 13277, 50697, 194327, 747319, 2882061, 11142027, 43167573, 167561586, 651513594, 2537041938, 9892847952, 38623197264, 150959213886, 590626854072, 2312979822738, 9065733950526, 35561306875380, 139595183125750

Offset: 0

Seiichi Manyama, Apr 10 2024

nonn

Cf. A360150, A371872, A371873.

Cf. A105872, A371758.

A371871 := proc(n)
    1/(1-x^3)/(1-x)^(n-1) ;
    coeftayl(%,x=0,n) ;
end proc:
seq(A371871(n),n=0..60) ; # R. J. Mathar, Apr 22 2024

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

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

D-finite with recurrence 9*n*a(n) +3*(-17*n+16)*a(n-1) +3*(21*n-50)*a(n-2) +(-17*n+16)*a(n-3) +10*(2*n-5)*a(n-4)=0. - R. J. Mathar, Apr 22 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

A371873 a(n) = Sum_{k=0..floor(n/3)} binomial(2n+1,n-3k).

Original entry on oeis.org

1, 3, 10, 36, 135, 517, 2003, 7815, 30634, 120480, 475002, 1876294, 7422676, 29400192, 116567356, 462561572, 1836843591, 7298613997, 29016050831, 115408159467, 459209330821, 1827849895817, 7277945888781, 28986847296997, 115479393316211, 460159673245743

Offset: 0

Seiichi Manyama, Apr 10 2024

nonn

Cf. A360150, A371871, A371872.

Cf. A002450, A360168, A371777.

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

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

A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2n+k,n-2k).

Original entry on oeis.org

1, 2, 7, 27, 107, 429, 1731, 7012, 28478, 115864, 471991, 1924483, 7852083, 32053208, 130893949, 534673600, 2184482707, 8926392419, 36479840422, 149095843951, 609400587426, 2490900041118, 10181669553847, 41618414303969, 170118507902985, 695366323719302

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

Cf. A014300, A106188, A108081, A114121, A176287.

Cf. A000108, A360144, A360150.

A360149 := proc(n)
    add(binomial(2*n+k,n-2*k),k=0..floor(n/2)) ;
end proc:
seq(A360149(n),n=0..40) ; # R. J. Mathar, Mar 02 2023

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

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

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

G.f.: 1 / ( sqrt(1-4*x) * (1 - x^2 * c(x)^5) ), where c(x) is the g.f. of A000108.
a(n) ~ sqrt((7 - 5*(2/(173 + 21*sqrt(69)))^(1/3) + ((173 + 21*sqrt(69))/2)^(1/3)) / 69) / ((4 - (2/(25 - 3*sqrt(69)))^(1/3) - ((25 - 3*sqrt(69))/2)^(1/3))/3)^n. - Vaclav Kotesovec, Jan 28 2023
D-finite with recurrence n*(47*n-1011)*a(n) +(-261*n^2 +8567*n -6378)*a(n-1) +2*(-165*n^2 -9388*n +16143)*a(n-2) +(3089*n^2 +919*n -27492)*a(n-3) +2*(-1283*n^2 +3900*n +3981)*a(n-4) +4*(81*n+11)*(2*n-9)*a(n-5)=0. - R. J. Mathar, Mar 02 2023

A371854 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-k+2,n-3k).

Original entry on oeis.org

1, 4, 15, 57, 219, 847, 3290, 12819, 50066, 195909, 767790, 3013002, 11837043, 46548919, 183209125, 721628692, 2844297119, 11217639757, 44265835891, 174765349896, 690308413773, 2727823240762, 10783518961394, 42644560775835, 168699835910561, 667580653569309

Offset: 0

Seiichi Manyama, Apr 09 2024

nonn

Cf. A002450, A105872, A371842.

Cf. A360150, A371773.

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

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

cp's OEIS Frontend

A105872 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k, n).

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

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A371871 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-3*k-2,n-3*k).

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

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A371873 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n+1,n-3*k).

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A105872 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-3k, n).

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).

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

A371871 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-3k-2,n-3*k).

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

A371873 a(n) = Sum_{k=0..floor(n/3)} binomial(2n+1,n-3k).

A360149 a(n) = Sum_{k=0..floor(n/2)} binomial(2n+k,n-2k).

A371854 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-k+2,n-3k).