A006139 -id:A006139 - cp's OEIS frontend

A103821 A Whitney transform of the central binomial coefficients A000984.

1, 3, 11, 43, 179, 771, 3395, 15171, 68515, 311907, 1428835, 6578531, 30414435, 141105251, 656588899, 3063038051, 14321092195, 67088405091, 314825048675, 1479654425187, 6963888239203, 32815960756835, 154813864252003

Offset: 0

Paul Barry, Feb 16 2005

Partial sums of A006139. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

CoefficientList[Series[1/((1-x)*Sqrt[1-4*x-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

G.f. : 1/((1-x)sqrt(1-4x-4x^2));
a(n)=sum{k=0..n, sum{i=0..n, C(k, i-k)}*C(2k, k)}.
Conjecture: n*a(n) +(2-5n)*a(n-1) +2*a(n-2)+4*(n-1)*a(n-3)=0. - R. J. Mathar, Dec 14 2011
a(n) ~ sqrt(34+23*sqrt(2))*(2+2*sqrt(2))^n/(7*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012

A206248 Number of (n+1) X 2 0..3 arrays with every 2 X 2 subblock having zero permanent.

Original entry on oeis.org

49, 361, 1600, 9409, 47089, 258064, 1343281, 7198489, 37945600, 201895681, 1068570721, 5672499856, 30061664689, 159465247561, 845443470400, 4483691905729, 23774527583569, 126075439502224, 668536407557041

Offset: 1

R. H. Hardin, Feb 05 2012

nonn

Column 1 of A206255.

			Some solutions for n=4:
..1..0....2..1....2..1....2..3....2..3....3..3....3..3....0..3....0..3....0..1
..0..0....0..0....0..0....0..0....0..0....0..0....0..0....0..2....0..2....0..0
..0..3....1..0....2..1....2..3....0..0....1..1....3..1....0..3....0..0....0..1
..0..1....2..0....0..0....0..0....0..0....0..0....0..0....0..2....3..3....0..2
..0..3....1..0....3..0....1..3....3..0....1..1....0..1....0..1....0..0....0..2

R. H. Hardin, Table of n, a(n) for n = 1..210

Cf. A206255.

Empirical: a(n) = 4*a(n-1) +12*a(n-2) -27*a(n-3).
a(n) = A006139(n+2)^2.
Conjectures from Colin Barker, Jun 14 2018: (Start)
G.f.: x*(49 + 165*x - 432*x^2) / ((1 + 3*x)*(1 - 7*x + 9*x^2)).
a(n) = 2^(-n)*((-1)^n*2^(1+n)*3^(3+n) + (77-20*sqrt(13))*(7-sqrt(13))^n + (7+sqrt(13))^n*(77+20*sqrt(13))) / 13.
(End)

A257365 Triangle, read by rows, T(n,k) = Sum_{m=0..(n-k)/2} C(k,m)C(n-2m,k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 8, 4, 1, 1, 8, 16, 13, 5, 1, 1, 10, 28, 32, 19, 6, 1, 1, 12, 44, 68, 55, 26, 7, 1, 1, 14, 64, 128, 136, 86, 34, 8, 1, 1, 16, 88, 220, 296, 241, 126, 43, 9, 1, 1, 18, 116, 352, 584, 592, 393, 176, 53, 10, 1

Offset: 0

Vladimir Kruchinin, Apr 21 2015

From Emanuele Munarini, Feb 21 2017: (Start)
T(n,k) is the number of lattice paths from (0,0) to (n,k) using steps X=(1,0), D=(1,1) and E=(3,1).
Row sums = A008998.
Central coefficients = A006139. (End)

			1;
1, 1;
1, 2, 1;
1, 4, 3, 1;
1, 6, 8, 4, 1;
1, 8, 16, 13, 5, 1;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened)
James East, Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.

Cf. A006139.

Table[Sum[Binomial[k, m] Binomial[n - 2 m, k], {m, 0, (n - k)/2}], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 21 2015 *)

T(n,k):=sum(binomial(k,m)*binomial(n-2*m,k),m,0,(n-k)/2);

G.f.: 1/(1-y-x*(1+y^2)).
From Emanuele Munarini, Feb 21 2017: (Start)
G.f. for the triangle: 1/(1-x-x*y-x^3*y).
Recurrence: T(n+3,k+1) = T(n+2,k+1) + T(n+2,k) + T(n,k). (End)

A343842 Series expansion of 1/sqrt(8*x^2 + 1), even powers only.

Original entry on oeis.org

1, -4, 24, -160, 1120, -8064, 59136, -439296, 3294720, -24893440, 189190144, -1444724736, 11076222976, -85201715200, 657270374400, -5082890895360, 39392404439040, -305870434467840, 2378992268083200, -18531097667174400, 144542561803960320, -1128808577897594880

Offset: 0

Peter Luschny, May 04 2021

Essentially the inverse binomial convolution of the Delannoy numbers.

Signed version of A059304.

Cf. A008288, A006139.

gf := 1/sqrt(8*x^2 + 1): ser := series(gf, x, 32):
seq(coeff(ser, x, 2*n), n = 0..21);

Take[CoefficientList[Series[1/Sqrt[8*x^2 + 1], {x, 0, 42}], x], {1, -1, 2}] (* Amiram Eldar, May 05 2021 *)

my(x='x+O('x^25)); Vec(1/sqrt(8*x + 1)) \\ Michel Marcus, May 04 2021

a(n) = n! * [x^n] BesselJ(0, sqrt(8)*x).
D-finite with recurrence a(n) = 4*(1 - 2*n)*a(n - 1) / n for n >= 2.
a(n) = A(2*n) where A(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*A008288(n, k).

A361488 Diagonal of rational function 1/(1 - (x^3 + y^3 + x^4*y)).

Original entry on oeis.org

1, 0, 0, 2, 2, 0, 6, 12, 6, 20, 60, 60, 90, 280, 420, 532, 1330, 2520, 3444, 6804, 14112, 21912, 37884, 77616, 133914, 223080, 432432, 793364, 1341912, 2471040, 4629196, 8076640, 14453010, 26960232, 48308832, 85794852, 157947816, 287413152, 512697900, 933072064

Offset: 0

Seiichi Manyama, Mar 22 2023

nonn

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

Cf. A006139, A115962.

Cf. A360267, A361727.

Table[Sum[Binomial[2*k,k] * Binomial[k,n-3*k], {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 23 2023 *)

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

G.f.: 1/sqrt(1 - 4 * x^3 * (1+x)).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(k,n-3*k).
From Vaclav Kotesovec, Mar 23 2023: (Start)
Recurrence: n*a(n) = 2*(2*n-3)*a(n-3) + 4*(n-2)*a(n-4).
a(n) ~ sqrt(c) * d^n / sqrt(Pi*n), where d = 1.835086681639635368143322042736678753... is the positive real root of the equation d^4 - 4*d - 4 = 0 and c = 0.2982650309662120181812121016104223... is the largest real root of the equation 1 - 20*c + 132*c^2 - 364*c^3 + 364*c^4 = 0. (End)

A370285 Coefficient of x^n in the expansion of ( (1+x)^2 + x^3 )^n.

Original entry on oeis.org

1, 2, 6, 23, 94, 392, 1659, 7107, 30734, 133880, 586576, 2582142, 11411371, 50597900, 224986467, 1002867878, 4479814606, 20049099908, 89878609344, 403521966942, 1814102538624, 8165526187128, 36794746597494, 165968135843522, 749314496125451, 3385881647958442

Offset: 0

Seiichi Manyama, Feb 14 2024

nonn

Cf. A006139, A026375, A369212.

Similar to A082759.

a := n -> binomial(2*n, n) * hypergeom([(1-n)/3, (2-n)/3, -n/3], [1/2-n, n+1], 27/4):
seq(simplify(a(n)), n = 0..25);  # Peter Luschny, Jan 04 2025

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

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^2 + x^3) ). See A369212.

A370286 Coefficient of x^n in the expansion of ( (1+x)^3 + x^2 )^n.

Original entry on oeis.org

1, 3, 17, 102, 645, 4193, 27764, 186231, 1261213, 8604759, 59053167, 407217396, 2819252544, 19583729766, 136426565999, 952743556907, 6667916884701, 46755146944959, 328398159653117, 2310073990369926, 16271915501598595, 114757849228310355

Offset: 0

Seiichi Manyama, Feb 14 2024

nonn

Cf. A002426, A006139.
Cf. A188686, A370287.
Cf. A065065.

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

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x / ((1+x)^3 + x^2) ).

A372086 G.f. A(x) satisfies A(x) = 1/( 1 - 4x(1 + x)*A(x) )^(1/2).

Original entry on oeis.org

1, 2, 12, 84, 664, 5624, 49928, 458480, 4318896, 41503168, 405267840, 4009684608, 40109536992, 404977089792, 4121810315584, 42243756499328, 435593319238144, 4515817056202496, 47040399766319360, 492117122988795392, 5168258741713256960

Offset: 0

Seiichi Manyama, Apr 17 2024

nonn

Cf. A006139, A372037.

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

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

cp's OEIS Frontend

A103821 A Whitney transform of the central binomial coefficients A000984.

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A206248 Number of (n+1) X 2 0..3 arrays with every 2 X 2 subblock having zero permanent.

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A257365 Triangle, read by rows, T(n,k) = Sum_{m=0..(n-k)/2} C(k,m)*C(n-2*m,k).

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Maxima

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A343842 Series expansion of 1/sqrt(8*x^2 + 1), even powers only.

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Maple

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PARI

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A361488 Diagonal of rational function 1/(1 - (x^3 + y^3 + x^4*y)).

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PARI

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A370285 Coefficient of x^n in the expansion of ( (1+x)^2 + x^3 )^n.

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Maple

PARI

Formula

A370286 Coefficient of x^n in the expansion of ( (1+x)^3 + x^2 )^n.

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PARI

Formula

A372086 G.f. A(x) satisfies A(x) = 1/( 1 - 4*x*(1 + x)*A(x) )^(1/2).

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A257365 Triangle, read by rows, T(n,k) = Sum_{m=0..(n-k)/2} C(k,m)C(n-2m,k).

A372086 G.f. A(x) satisfies A(x) = 1/( 1 - 4x(1 + x)*A(x) )^(1/2).