A105872 -id:A105872 - cp's OEIS frontend

A176290 Hankel transform of A105872.

1, 2, -3, -75, -650, -4507, -28267, -167406, -955271, -5310911, -28962586, -155616567, -826329687, -4345964510, -22675946635, -117526104883, -605643805098, -3105646720979, -15856669574339, -80653146223054

Offset: 0

Paul Barry, Apr 14 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (10,-27,10,-1).

Cf. A105872.

a:=[1,2,-3,-75];; for n in [5..30] do a[n]:=10*a[n-1]-27*a[n-2]+10*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Nov 25 2019

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-8*x+4*x^2-x^3)/(1-5*x+x^2)^2 )); // G. C. Greubel, Nov 25 2019

seq(coeff(series((1-8*x+4*x^2-x^3)/(1-5*x+x^2)^2, x, n+1), x, n), n = 0..30); # G. C. Greubel, Nov 25 2019

LinearRecurrence[{10,-27,10,-1},{1,2,-3,-75},30] (* Harvey P. Dale, Oct 29 2017 *)

my(x='x+O('x^30)); Vec((1-8*x+4*x^2-x^3)/(1-5*x+x^2)^2) \\ G. C. Greubel, Nov 25 2019

def A176290_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-8*x+4*x^2-x^3)/(1-5*x+x^2)^2).list()
A176290_list(30) # G. C. Greubel, Nov 25 2019

G.f.: (1-8*x+4*x^2-x^3)/(1-5*x+x^2)^2.

A360150 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-k,n-3k).

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Jan 28 2023

nonn

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

Cf. A000108.

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

a[n_] := Sum[Binomial[2*n - 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-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)))^5)))

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

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

A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

Original entry on oeis.org

1, 2, 6, 19, 65, 231, 841, 3110, 11628, 43834, 166298, 634140, 2428336, 9331688, 35967462, 138987715, 538287881, 2088842463, 8119916647, 31613327405, 123251518641, 481125828853, 1880262896537, 7355767408395, 28803717914791, 112887697489907, 442784607413427

Offset: 0

Seiichi Manyama, Apr 03 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1664

Partial sums of A026641. - Seiichi Manyama, Jan 30 2023
Cf. A000108, A006134, A079309, A105872, A120305, A306409.
Cf. A191993, A360186, A360212.

Table[Sum[Sum[(-1)^(i + j)*(i + j)!/(i!*j!), {i, 0, j}], {j, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 04 2019 *)

a(n) = sum(i=0, n, sum(j=i, n, (-1)^(i+j)*(i+j)!/(i!*j!)));

a(n) = sum(k=0, n\3, (-1)^k*binomial(2*n-3*k, n)); \\ Seiichi Manyama, Jan 29 2023

my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^3*(2/(1+sqrt(1-4*x)))^3))) \\ Seiichi Manyama, Jan 29 2023

a(n) = (A006134(n) + A120305(n))/2.
From Vaclav Kotesovec, Apr 04 2019: (Start)
Recurrence: 2*n*a(n) = (9*n-4)*a(n-1) - (3*n-2)*a(n-2) - 2*(2*n-1)*a(n-3).
a(n) ~ 2^(2*n+3) / (9*sqrt(Pi*n)). (End)
From Seiichi Manyama, Jan 29 2023: (Start)
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k,n).
G.f.: 1 / ( sqrt(1-4*x) * (1 + x^3 * c(x)^3) ), where c(x) is the g.f. of A000108. (End)
a(n) = [x^n] 1/((1+x^3) * (1-x)^(n+1)). - Seiichi Manyama, Apr 08 2024

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

A371842 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-2k+1,n-3*k).

Original entry on oeis.org

1, 3, 10, 36, 133, 498, 1882, 7161, 27391, 105210, 405499, 1567332, 6072724, 23578221, 91712089, 357301827, 1393986898, 5445422340, 21296030401, 83370591273, 326688422203, 1281227165640, 5028742763407, 19751799462378, 77632592859316, 305316702610581

Offset: 0

Seiichi Manyama, Apr 08 2024

nonn

Cf. A105872.

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)).
From Vaclav Kotesovec, Apr 08 2024: (Start)
Recurrence: (n-1)*a(n) = (9*n-11)*a(n-1) - 2*(11*n-16)*a(n-2) + (9*n-13)*a(n-3) - 2*(2*n-3)*a(n-4).
G.f.: 2 / (4*x^2 + 3*x*sqrt(1-4*x) - 9*x + 2).
a(n) ~ 2^(2*n+3) / (3*sqrt(Pi*n)). (End)

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

A176290 Hankel transform of A105872.

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A360150 a(n) = Sum_{k=0..floor(n/3)} binomial(2*n-k,n-3*k).

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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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A307354 a(n) = Sum_{0<=i<=j<=n} (-1)^(i+j) * (i+j)!/(i!*j!).

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

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A360150 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-k,n-3k).

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).

A371842 a(n) = Sum_{k=0..floor(n/3)} binomial(2n-2k+1,n-3*k).

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