A364339 -id:A364339 - cp's OEIS frontend

A364336 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^3).

1, 2, 7, 39, 242, 1634, 11631, 85957, 653245, 5072862, 40077807, 321106623, 2602911282, 21308131235, 175909559897, 1462846379247, 12242600576066, 103035285071630, 871490142773640, 7404121610615520, 63157400073057627, 540689217572662413, 4644083121177225292

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

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

Cf. A073157, A364337, A364338, A364339.

Cf. A198953, A212071, A215623, A215654, A216359.

A364336 := proc(n)
    add( binomial(3*k+1,k) * binomial(3*k+1,n-k)/(3*k+1),k=0..n) ;
end proc:
seq(A364336(n),n=0..80); # R. J. Mathar, Jul 25 2023

nmax = 80; A[_] = 1;
Do[A[x_] = (1 + x)*(1 + x*A[x]^3) + O[x]^(nmax+1) // Normal, {nmax+1}];
CoefficientList[A[x], x] (* Jean-François Alcover, Mar 03 2024 *)

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

a(n) = Sum_{k=0..n} binomial(3*k+1,k) * binomial(3*k+1,n-k) / (3*k+1).
D-finite with recurrence -2*n*(2*n+1)*a(n) +(3*n^2+23*n-14)*a(n-1) +(207*n^2 -635*n +494)*a(n-2) +2*(397*n^2 -2031*n +2600)*a(n-3) +6*(75*n-244) *(3*n-11)*a(n-4) +9*(45*n-179) *(3*n-14)*a(n-5) +63*(3*n-14) *(3*n-17)*a(n-6) +12*(3*n-16) *(3*n-20)*a(n-7)=0. - R. J. Mathar, Jul 25 2023
From Peter Bala, Sep 10 2024: (Start)
x/series_reversion(x*A(x)) = 1 + 2*x + 3*x^2 + 13*x^3 + 32*x^4 + 147*x^5 + ..., the g.f. of A216359.
(1/x) * series_reversion(x/A(x)) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + 8551*x^5 + ..., the g.f. of A215623. (End)

A364337 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^4).

Original entry on oeis.org

1, 2, 9, 68, 580, 5406, 53270, 545844, 5757332, 62094217, 681653493, 7591431752, 85558696024, 974024788280, 11184192097016, 129378232148016, 1506363564912368, 17639001584452320, 207593804132718948, 2454236122156830254, 29132714097692056954, 347086786035103983446

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..907

Cf. A073157, A364336, A364338, A364339.

Cf. A215623, A215715, A234461, A239107.

terms = 22; A[] = 0; Do[A[x] = (1+x)(1+x*A[x]^4) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 24 2025 *)

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

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

A364338 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 11, 105, 1140, 13555, 170637, 2235472, 30161255, 416248640, 5848462880, 83378361111, 1203100853951, 17537182300140, 257858115407535, 3819894878557990, 56958234329850060, 854192593184162160, 12875579347191388830, 194963091634569681550, 2964229359714424159370, 45234864131654311730160

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..827

Cf. A073157, A364336, A364337, A364339.

Cf. A215624, A234525, A239108, A364331, A364335.

terms = 22; A[] = 0; Do[A[x] = (1+x)(1+x*A[x]^5) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 24 2025 *)

a(n) = sum(k=0, n, binomial(5*k+1, k)*binomial(5*k+1, n-k)/(5*k+1));

a(n) = Sum_{k=0..n} binomial(5*k+1,k) * binomial(5*k+1,n-k) / (5*k+1).

A366326 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^2).

Original entry on oeis.org

1, 2, -3, 14, -78, 479, -3131, 21372, -150588, 1087057, -7998295, 59763129, -452257495, 3459109408, -26697940390, 207672518808, -1626400971710, 12813379464399, -101482102525511, 807524595076284, -6452856224076654, 51760509258982478, -416620859045829372

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Winston de Greef, Table of n, a(n) for n = 0..1067

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366327, A366328.

Cf. A006013, A364393.

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

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

A364333 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^6).

Original entry on oeis.org

1, 2, 17, 216, 3224, 52640, 910452, 16392140, 303996224, 5767278431, 111401778266, 2183535060362, 43319505976084, 868220464851417, 17552981176788200, 357544690982030744, 7330803752675100908, 151172599088871911072, 3133367418601958989295, 65242183918761533467216

Offset: 0

Seiichi Manyama, Jul 18 2023

nonn

Cf. A007863, A069271, A073157, A215654, A215715, A364331.
Cf. A239109, A364339, A364340.
Cf. A200718.

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

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

A366325 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)).

Original entry on oeis.org

1, 2, -1, 3, -10, 36, -137, 543, -2219, 9285, -39587, 171369, -751236, 3328218, -14878455, 67030785, -304036170, 1387247580, -6363044315, 29323149825, -135700543190, 630375241380, -2938391049395, 13739779184085, -64430797069375, 302934667061301, -1427763630578197

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Cf. A073157, A007863, A364336, A364337, A364338, A364339, A366326, A366327, A366328.

Cf. A000108, A002212, A112478.

a := proc(n) option remember; if n = 1 then 2 elif n = 2 then -1 else (-3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2))/n fi; end:
seq(a(n), n = 1..30); # Peter Bala, Sep 10 2024

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

G.f.: A(x) = -2*x*(1+x) / (1+x-sqrt((1+x)*(1+5*x))).
a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(n-2,n-k)/(2*k-1).
a(n) ~ -(-1)^n * 5^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 07 2023
From Peter Bala, Sep 10 2024: (Start)
a(n) = 1/(1 - n) * Sum_{k = 0..n} binomial(-n+k, k)*binomial(-n+k+1, n-k) for n not equal to 1. Cf. A007863.
a(n) = Sum_{k = 0..n-2} binomial(-n+k+1, k)*binomial(-n+k+1, n-k)/(-n+k+1) for n >= 2.
P-recursive: n*a(n) = - 3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2) with a(1) = 2 and a(2)  = -1. (End)

A366327 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^3).

Original entry on oeis.org

1, 2, -5, 33, -260, 2263, -20979, 203124, -2030121, 20786694, -216928144, 2298911699, -24673591005, 267644087524, -2929602893537, 32317666058508, -358931896710948, 4010200327457883, -45040693394259858, 508253687784232108, -5759468659295939684

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366326, A366328.

Cf. A006632, A364407.

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

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

A366328 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^4).

Original entry on oeis.org

1, 2, -7, 60, -612, 6898, -82806, 1038076, -13431940, 178040315, -2405137161, 32992706368, -458336721104, 6435090557964, -91167680664004, 1301665779507128, -18710805300530504, 270559054510943509, -3932893180646204203, 57437414168562779574, -842365843304975785062

Offset: 0

Seiichi Manyama, Oct 07 2023

sign

Cf. A073157, A364336, A364337, A364338, A364339, A366325, A366326, A366327.

Cf. A118971, A364408.

a(n) = (-1)^(n-1)*sum(k=0, n, binomial(5*k-1, k)*binomial(n+3*k-2, n-k)/(5*k-1));

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(5*k-1,k) * binomial(n+3*k-2,n-k)/(5*k-1).

A364340 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^6).

Original entry on oeis.org

1, 2, 15, 179, 2502, 38262, 619991, 10459410, 181771289, 3231782239, 58505593456, 1074766446526, 19984671314164, 375414901633692, 7113886504446443, 135820770971898805, 2610186429457347486, 50452256583633573513, 980187901557594671335, 19130197594133100828170, 374894511736219913097375

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Cf. A239109, A364333, A364339.

Cf. A007863, A198953, A215623, A215624.

a(n) = sum(k=0, n, binomial(n+5*k+1, k)*binomial(n+5*k+1, n-k)/(n+5*k+1));

a(n) = Sum_{k=0..n} binomial(n+5*k+1,k) * binomial(n+5*k+1,n-k) / (n+5*k+1).

cp's OEIS Frontend

A364336 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^3).

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A364337 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^4).

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A364338 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^5).

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A366326 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^2).

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A364333 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^6).

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A366325 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)).

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Maple

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A366327 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^3).

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A366328 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)^4).

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A364340 G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^6).

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A364333 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^6).

A364340 G.f. satisfies A(x) = (1 + xA(x)) (1 + x*A(x)^6).