A361812 -id:A361812 - cp's OEIS frontend

A361816 Expansion of 1/sqrt(1 - 4x(1-x)^3).

1, 2, 0, -10, -22, 12, 174, 344, -354, -3304, -5780, 9180, 65258, 99132, -226620, -1313580, -1690990, 5441340, 26681700, 28070100, -128211552, -543818824, -440381780, 2978145240, 11080939914, 6162798092, -68377892976, -225107280388, -64286124152

Offset: 0

Seiichi Manyama, Mar 25 2023

sign

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

Column k=3 of A361834.
Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361815, A361817.
Cf. A361812.

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x)^3))

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

n*a(n) = 2 * ( (2*n-1)*a(n-1) - 3*(2*n-2)*a(n-2) + 3*(2*n-3)*a(n-3) - (2*n-4)*a(n-4) ) for n > 3.

A361813 Expansion of 1/sqrt(1 - 4x(1+x)^4).

Original entry on oeis.org

1, 2, 14, 80, 486, 3030, 19184, 122924, 794678, 5173160, 33863666, 222683588, 1469908848, 9733916596, 64636957300, 430240178484, 2869778018070, 19177245746844, 128361805431752, 860443079597872, 5775392952659170, 38811408514848032, 261101034656317244

Offset: 0

Seiichi Manyama, Mar 25 2023

nonn

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361812, A361814.

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

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

n*a(n) = 2 * ( (2*n-1)*a(n-1) + 4*(2*n-2)*a(n-2) + 6*(2*n-3)*a(n-3) + 4*(2*n-4)*a(n-4) + (2*n-5)*a(n-5) ) for n > 4.

A361814 Expansion of 1/sqrt(1 - 4x(1+x)^5).

Original entry on oeis.org

1, 2, 16, 100, 660, 4482, 30886, 215364, 1515000, 10730800, 76426846, 546792056, 3926775646, 28290272420, 204375145480, 1479963148220, 10739326203132, 78072933869364, 568503202324540, 4145718464390120, 30271771382355430, 221305746414518180

Offset: 0

Seiichi Manyama, Mar 25 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..1139

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361812, A361813.

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

a(n)= sum(k=0, n, binomial(2*k,k) * binomial(5*k,n-k)) \\ Winston de Greef, Mar 25 2023

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(5*k,n-k).

n*a(n) = 2 * ( (2*n-1)*a(n-1) + 5*(2*n-2)*a(n-2) + 10*(2*n-3)*a(n-3) + 10*(2*n-4)*a(n-4) + 5*(2*n-5)*a(n-5) + (2*n-6)*a(n-6) ) for n > 5.

A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2j,j) binomial(k*j,n-j).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 8, 20, 1, 2, 10, 32, 70, 1, 2, 12, 46, 136, 252, 1, 2, 14, 62, 226, 592, 924, 1, 2, 16, 80, 342, 1136, 2624, 3432, 1, 2, 18, 100, 486, 1932, 5810, 11776, 12870, 1, 2, 20, 122, 660, 3030, 11094, 30080, 53344, 48620

Offset: 0

Seiichi Manyama, Mar 26 2023

			Square array begins:
    1,   1,    1,    1,    1,    1, ...
    2,   2,    2,    2,    2,    2, ...
    6,   8,   10,   12,   14,   16, ...
   20,  32,   46,   62,   80,  100, ...
   70, 136,  226,  342,  486,  660, ...
  252, 592, 1136, 1932, 3030, 4482, ...

Columns k=0..5 give A000984, A006139, A137635, A361812, A361813, A361814.
Main diagonal gives A361829.
Cf. A099233, A361834.

T(n, k) = sum(j=0, n, binomial(2*j, j)*binomial(k*j, n-j));

G.f. of column k: 1/sqrt(1 - 4*x*(1+x)^k).

n*T(n,k) = 2 * Sum_{j=0..k} binomial(k,j)*(2*n-1-j)*T(n-1-j,k) for n > k.

A361842 Expansion of 1/(1 - 9x(1+x)^3)^(1/3).

Original entry on oeis.org

1, 3, 27, 243, 2352, 23607, 242757, 2539431, 26904492, 287858421, 3104029755, 33684914907, 367483636746, 4026930734223, 44295829667055, 488855016668727, 5410588668898995, 60035381850523284, 667643481187840206, 7439651232903588528, 83050643822779921347

Offset: 0

Seiichi Manyama, Mar 26 2023

nonn

Winston de Greef, Table of n, a(n) for n = 0..932

Column k=3 of A361839.

Cf. A099234, A361812, A361845.

a[n_]:=(-9)^n*Binomial[-1/3, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 2/3-n, 2/3-n}, -2^8/3^5]; Array[a,21,0] (* Stefano Spezia, Jul 11 2024 *)

my(N=30, x='x+O('x^N)); Vec(1/(1-9*x*(1+x)^3)^(1/3))

n*a(n) = 3 * ( (3*n-2)*a(n-1) + 3*(3*n-4)*a(n-2) + 3*(3*n-6)*a(n-3) + (3*n-8)*a(n-4) ) for n > 3.
a(n) = Sum_{k=0..n} (-9)^k * binomial(-1/3,k) * binomial(3*k,n-k).
a(n) = (-9)^n*binomial(-1/3, n)*hypergeom([(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4], [1/3-n, 2/3-n, 2/3-n], -2^8/3^5). - Stefano Spezia, Jul 11 2024

cp's OEIS Frontend

A361816 Expansion of 1/sqrt(1 - 4*x*(1-x)^3).

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A361813 Expansion of 1/sqrt(1 - 4*x*(1+x)^4).

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A361814 Expansion of 1/sqrt(1 - 4*x*(1+x)^5).

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A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2*j,j) * binomial(k*j,n-j).

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A361842 Expansion of 1/(1 - 9*x*(1+x)^3)^(1/3).

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Mathematica

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A361816 Expansion of 1/sqrt(1 - 4x(1-x)^3).

A361813 Expansion of 1/sqrt(1 - 4x(1+x)^4).

A361814 Expansion of 1/sqrt(1 - 4x(1+x)^5).

A361830 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(2j,j) binomial(k*j,n-j).

A361842 Expansion of 1/(1 - 9x(1+x)^3)^(1/3).