A364338 -id:A364338 - 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).

A364339 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^6).

Original entry on oeis.org

1, 2, 13, 150, 1978, 28603, 438273, 6992052, 114915180, 1932233883, 33081722359, 574755965137, 10107627041697, 179576579730534, 3218352405778284, 58114340679967608, 1056284029850962674, 19310039426151335622, 354818596435147647654, 6549556302551204621664, 121394125733645986376838

Offset: 0

Seiichi Manyama, Jul 19 2023

nonn

Cf. A073157, A364336, A364337, A364338.

Cf. A239109, A364333, A364340.

terms = 21; A[] = 0; Do[A[x] = (1+x)(1+x*A[x]^6) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 24 2025 *)

a(n) = sum(k=0, n, binomial(6*k+1, k)*binomial(6*k+1, n-k)/(6*k+1));

a(n) = Sum_{k=0..n} binomial(6*k+1,k) * binomial(6*k+1,n-k) / (6*k+1).

A364331 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 15, 163, 2070, 28698, 421015, 6425644, 100977137, 1622885389, 26551709946, 440744175801, 7404449354076, 125657625548824, 2150963575012295, 37094953102567208, 643904274979347286, 11241232087809137759, 197247501440314516840, 3476787208220672891388, 61533794803235280779261

Offset: 0

Seiichi Manyama, Jul 18 2023

Cf. A007863, A069271, A073157, A215654, A215715, A364333.
Cf. A215623, A215624, A239108, A364335, A364338.
Cf. A200719.

A364331 := proc(n)
    add( binomial(2*n+3*k+1,k) * binomial(2*n+3*k+1,n-k)/(2*n+3*k+1),k=0..n) ;
end proc:
seq(A364331(n),n=0..70); # R. J. Mathar, Jul 25 2023

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

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

x/series_reversion(x*A(x)) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + ..., the g.f. of A215623. - Peter Bala, Sep 08 2024

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).

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).

A364335 G.f. satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 17, 204, 2852, 43489, 701438, 11767095, 203223146, 3589167533, 64524575635, 1176860764416, 21723084076739, 405038036077647, 7617437252889030, 144328483391622298, 2752414654270742784, 52790626691557217602, 1017655117382823639414, 19706520281177438174530

Offset: 0

Seiichi Manyama, Jul 18 2023

nonn

Cf. A215624, A234525, A239108, A364331, A364338.

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

a(n) = Sum_{k=0..n} binomial(3*n+2*k+1,k) * binomial(3*n+2*k+1,n-k) / (3*n+2*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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A364339 G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^6).

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A364331 G.f. satisfies A(x) = (1 + x*A(x)^2) * (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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A366325 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)).

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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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A364335 G.f. satisfies A(x) = (1 + x*A(x)^3) * (1 + x*A(x)^5).

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A364331 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^5).

A364335 G.f. satisfies A(x) = (1 + xA(x)^3) (1 + x*A(x)^5).