A361830 -id:A361830 - cp's OEIS frontend

A361812 Expansion of 1/sqrt(1 - 4x(1+x)^3).

1, 2, 12, 62, 342, 1932, 11094, 64480, 378150, 2233304, 13263772, 79136844, 473969586, 2847911596, 17159547804, 103640073972, 627280131594, 3803643145596, 23102172930156, 140522319418164, 855880464524472, 5219168576004184, 31861229045809436

Offset: 0

Seiichi Manyama, Mar 25 2023

nonn

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

Column k=3 of A361830.

Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361813, A361814.

a[n_]:=Binomial[2*n, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 1/2-n, 2/3-n}, -2^6/3^3]; Array[a,23,0] (* Stefano Spezia, Jul 11 2024 *)

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

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

A361834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2j,j) binomial(k*j,n-j).

Original entry on oeis.org

1, 1, 2, 1, 2, 6, 1, 2, 4, 20, 1, 2, 2, 8, 70, 1, 2, 0, -2, 16, 252, 1, 2, -2, -10, -14, 32, 924, 1, 2, -4, -16, -22, -32, 64, 3432, 1, 2, -6, -20, -10, 12, -30, 128, 12870, 1, 2, -8, -22, 20, 118, 174, 64, 256, 48620, 1, 2, -10, -22, 66, 242, 304, 344, 346, 512, 184756

Offset: 0

Seiichi Manyama, Mar 26 2023

			Square array begins:
    1,  1,   1,   1,   1,   1,    1, ...
    2,  2,   2,   2,   2,   2,    2, ...
    6,  4,   2,   0,  -2,  -4,   -6, ...
   20,  8,  -2, -10, -16, -20,  -22, ...
   70, 16, -14, -22, -10,  20,   66, ...
  252, 32, -32,  12, 118, 242,  342, ...
  924, 64, -30, 174, 304,  82, -678, ...

Columns k=0..4 give A000984, A000079, A361815, A361816, A361817.
Main diagonal gives A361835.
Cf. A361830.

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

A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9x(1 + x)^k)^(1/3).

Original entry on oeis.org

1, 1, 3, 1, 3, 18, 1, 3, 21, 126, 1, 3, 24, 162, 945, 1, 3, 27, 201, 1341, 7371, 1, 3, 30, 243, 1809, 11529, 58968, 1, 3, 33, 288, 2352, 16893, 101619, 480168, 1, 3, 36, 336, 2973, 23607, 161676, 911466, 3961386, 1, 3, 39, 387, 3675, 31818, 242757, 1574289, 8281737, 33011550

Offset: 0

Seiichi Manyama, Mar 26 2023

			Square array begins:
     1,     1,     1,     1,     1,     1, ...
     3,     3,     3,     3,     3,     3, ...
    18,    21,    24,    27,    30,    33, ...
   126,   162,   201,   243,   288,   336, ...
   945,  1341,  1809,  2352,  2973,  3675, ...
  7371, 11529, 16893, 23607, 31818, 41676, ...

Columns k=0..3 give A004987, A180400, A361841, A361842.
Main diagonal gives A361846.
Cf. A361830, A361840.

T(n,k) = sum(j=0, n, (-9)^j*binomial(-1/3, j)*binomial(k*j, n-j));

n*T(n,k) = 3 * Sum_{j=0..k} binomial(k,j)*(3*n-2-2*j)*T(n-1-j,k) for n > k.

T(n,k) = Sum_{j=0..n} (-9)^j * binomial(-1/3,j) * binomial(k*j,n-j).

A361829 a(n) = Sum_{k=0..n} binomial(2k,k) binomial(n*k,n-k).

Original entry on oeis.org

1, 2, 10, 62, 486, 4482, 47106, 553226, 7152438, 100644194, 1527758136, 24839853326, 430045385424, 7888706328934, 152685931935634, 3106864307092950, 66253232332628166, 1476558925897693698, 34307420366092350048, 829217371825336147142

Offset: 0

Seiichi Manyama, Mar 26 2023

nonn

Main diagonal of A361830.

Cf. A099237, A361835.

Table[Sum[Binomial[2*k,k]*Binomial[n*k,n-k], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 26 2023 *)

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

a(n) = [x^n] 1/sqrt(1 - 4*x*(1+x)^n).

log(a(n)) ~ n*(log(n) + (2*log(2) - 1)/log(n) - (1 - 1/log(n))*log(log(n) - 1)). - Vaclav Kotesovec, Mar 26 2023

cp's OEIS Frontend

A361812 Expansion of 1/sqrt(1 - 4x(1+x)^3).

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A361834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2j,j) binomial(k*j,n-j).

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A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9x(1 + x)^k)^(1/3).

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A361829 a(n) = Sum_{k=0..n} binomial(2k,k) binomial(n*k,n-k).

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cp's OEIS Frontend

A361812 Expansion of 1/sqrt(1 - 4*x*(1+x)^3).

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Author

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Mathematica

PARI

Formula

A361834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2*j,j) * binomial(k*j,n-j).

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A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 + x)^k)^(1/3).

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PARI

Formula

A361829 a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n*k,n-k).

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Formula

A361812 Expansion of 1/sqrt(1 - 4x(1+x)^3).

A361834 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (-1)^(n-j) * binomial(2j,j) binomial(k*j,n-j).

A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9x(1 + x)^k)^(1/3).

A361829 a(n) = Sum_{k=0..n} binomial(2k,k) binomial(n*k,n-k).