A366266 -id:A366266 - cp's OEIS frontend

A366455 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(5/2).

1, 2, -5, 30, -215, 1710, -14516, 128830, -1180920, 11093830, -106245975, 1033454774, -10181848705, 101394979530, -1018972470275, 10320779179380, -105250097458410, 1079767027094630, -11136159773691830, 115395278542757580, -1200814926210284360

Offset: 0

Seiichi Manyama, Oct 10 2023

sign

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366453, A366454, A366456.

Cf. A366401.

a(n) = (-1)^(n-1)*sum(k=0, n, binomial(7*k/2-1, k)*binomial(n+5*k/2-2, n-k)/(7*k/2-1));

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366401.

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(7*k/2-1,k) * binomial(n+5*k/2-2,n-k) / (7*k/2-1).

A366456 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(7/2).

Original entry on oeis.org

1, 2, -7, 56, -532, 5600, -62860, 737324, -8929726, 110811344, -1401640814, 18004922936, -234243536436, 3080152906096, -40870739065996, 546563064528906, -7358930622768977, 99672580921800656, -1357142384455626909, 18565841939010374736, -255054402946387767408

Offset: 0

Seiichi Manyama, Oct 10 2023

sign

Cf. A112478, A364393, A364407, A364408, A364409, A366266, A366267, A366268, A366452, A366453, A366454, A366455.

Cf. A366402.

a(n) = (-1)^(n-1)*sum(k=0, n, binomial(9*k/2-1, k)*binomial(n+7*k/2-2, n-k)/(9*k/2-1));

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366402.

a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(9*k/2-1,k) * binomial(n+7*k/2-2,n-k) / (9*k/2-1).

A366556 G.f. A(x) satisfies A(x) = 1 + x + x^4*A(x)^3.

Original entry on oeis.org

1, 1, 0, 0, 1, 3, 3, 1, 3, 15, 30, 30, 27, 87, 252, 420, 475, 747, 2064, 4632, 7203, 9933, 19635, 47025, 92013, 144745, 237510, 498498, 1073817, 1969131, 3267411, 5977881, 12462579, 25035747, 45090936, 79414344, 153115299, 311198457, 600883569, 1090988379, 2012793705

Offset: 0

Seiichi Manyama, Oct 13 2023

nonn

Cf. A019497, A366266, A366555.
Cf. A366554, A366558.
Cf. A366592.

a(n) = sum(k=0, n\4, binomial(2*k+1, n-4*k)*binomial(3*k, k)/(2*k+1));

a(n) = Sum_{k=0..floor(n/4)} binomial(2*k+1,n-4*k) * binomial(3*k,k)/(2*k+1).

a(n) = A366592(n) + A366592(n-1).

A348957 G.f. A(x) satisfies A(x) = (1 + x * A(-x)) / (1 - x * A(x)).

Original entry on oeis.org

1, 2, 2, 10, 18, 98, 210, 1194, 2786, 16258, 39906, 236938, 601458, 3615330, 9399858, 57024426, 150947010, 922283522, 2475603138, 15212318730, 41290579410, 254909413218, 698230131858, 4327273358250, 11943274468770, 74260741616514, 206279837823650, 1286199407132554

Offset: 0

Ilya Gutkovskiy, Nov 05 2021

nonn

Cf. A006318, A086246.
Cf. A349310, A361638, A364407, A366266, A366364.
Cf. A112478, A364394, A364395, A366363.

nmax = 27; A[] = 0; Do[A[x] = (1 + x A[-x])/(1 - x A[x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -(-1)^n a[n - 1] + Sum[a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 27}]
CoefficientList[y/.AsymptoticSolve[y-y^2+x(1+y^3)==0,y->1,{x,0,27}][[1]],x] (* Alexander Burstein, Nov 26 2021 *)

a(0) = 1; a(n) = -(-1)^n * a(n-1) + Sum_{k=0..n-1} a(k) * a(n-k-1).
a(n) ~ c * 3^(3*n/4) * (1 + sqrt(3))^n / (sqrt(2*Pi) * n^(3/2) * 2^(n/2)), where c = 3^(1/4) if n is even and c = (1 + sqrt(3))/sqrt(2) if n is odd. - Vaclav Kotesovec, Nov 14 2021
From Alexander Burstein, Nov 26 2021: (Start)
G.f.: A(-x) = 1/A(x).
G.f.: A(x) = 1 + x*(1+A(x)^3)/A(x). (End)
a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n-3*k-2,n-1) for n > 0. - Seiichi Manyama, Apr 11 2024

A378318 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3r+k,n)/(3r+k) for k > 0.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 6, 0, 1, 6, 16, 30, 0, 1, 8, 30, 84, 170, 0, 1, 10, 48, 170, 496, 1050, 0, 1, 12, 70, 296, 1050, 3140, 6846, 0, 1, 14, 96, 470, 1920, 6846, 20832, 46374, 0, 1, 16, 126, 700, 3210, 12936, 46374, 142932, 323154, 0, 1, 18, 160, 994, 5040, 22402, 89712, 323154, 1005856, 2301618, 0

Offset: 0

Seiichi Manyama, Nov 23 2024

			Square array begins:
  1,    1,     1,     1,     1,      1,      1, ...
  0,    2,     4,     6,     8,     10,     12, ...
  0,    6,    16,    30,    48,     70,     96, ...
  0,   30,    84,   170,   296,    470,    700, ...
  0,  170,   496,  1050,  1920,   3210,   5040, ...
  0, 1050,  3140,  6846, 12936,  22402,  36492, ...
  0, 6846, 20832, 46374, 89712, 159390, 266800, ...

Columns k=0..1 give A000007, A366266.
Main diagonal gives A378378.
Cf. A266213, A378317.
Cf. A378323.

T(n, k, t=0, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(n, r)*binomial(t*n+u*r+k, n)/(t*n+u*r+k)));
matrix(7, 7, n, k, T(n-1, k-1))

G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x * A_k(x)^(3/k) )^k for k > 0.
G.f. of column k: B(x)^k where B(x) is the g.f. of A366266.
B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k+2) for n > 0.

A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

Original entry on oeis.org

1, 1, 4, 15, 70, 360, 1953, 11008, 63837, 378390, 2282205, 13960890, 86411232, 540166219, 3405341160, 21625820793, 138216775785, 888371346825, 5738510504979, 37234351046835, 242567430368298, 1585979835198675, 10403866383915844, 68453912880893025

Offset: 0

Seiichi Manyama, Nov 03 2023

nonn

Cf. A176697, A367041.
Cf. A366266, A366676.
Cf. A002293, A104979, A186997, A255673, A361245, A364474, A364475, A364478.

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

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

A366696 G.f. satisfies A(x) = (1 + x)^2 + x*A(x)^3.

Original entry on oeis.org

1, 3, 10, 57, 378, 2730, 20853, 165592, 1353297, 11307168, 96148149, 829336122, 7238765532, 63816716547, 567425771478, 5082596905629, 45820260590481, 415423374916503, 3785371205061825, 34647928319586375, 318419608552433190, 2937021429784279707

Offset: 0

Seiichi Manyama, Oct 16 2023

nonn

Cf. A366266, A366697.

Cf. A364620.

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

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

A366697 G.f. satisfies A(x) = (1 + x)^3 + x*A(x)^3.

Original entry on oeis.org

1, 4, 15, 94, 706, 5769, 49923, 449376, 4164228, 39459852, 380594767, 3724049805, 36876008673, 368835076813, 3720863181033, 37815675159285, 386818379566749, 3979362306753315, 41144521893563511, 427335033811660713, 4456402044181677264

Offset: 0

Seiichi Manyama, Oct 16 2023

nonn

Cf. A366266, A366696.

Cf. A364623.

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

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

cp's OEIS Frontend

A366455 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(5/2).

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A366456 G.f. A(x) satisfies A(x) = 1 + x + x/A(x)^(7/2).

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A366556 G.f. A(x) satisfies A(x) = 1 + x + x^4*A(x)^3.

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A348957 G.f. A(x) satisfies A(x) = (1 + x * A(-x)) / (1 - x * A(x)).

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Mathematica

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A378318 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3*r+k,n)/(3*r+k) for k > 0.

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A367040 G.f. satisfies A(x) = 1 + x^2 + x*A(x)^3.

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A366696 G.f. satisfies A(x) = (1 + x)^2 + x*A(x)^3.

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A366697 G.f. satisfies A(x) = (1 + x)^3 + x*A(x)^3.

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A378318 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(n,r) * binomial(3r+k,n)/(3r+k) for k > 0.