A374506 -id:A374506 - cp's OEIS frontend

A245551 Expansion of 1/(1 - 2x - 3x^2)^(5/2).

1, 5, 25, 105, 420, 1596, 5880, 21120, 74415, 258115, 883883, 2994355, 10051860, 33479460, 110750580, 364177332, 1191186855, 3877914915, 12571302975, 40598200335, 130657125984, 419173385400, 1340928798300, 4278305877300, 13617034683525, 43243221276801, 137040737988105

Offset: 0

N. J. A. Sloane, Jul 30 2014

From Petros Hadjicostas, Jun 03 2020: (Start)
For n >= 4, 2*a(n-4) counts 3-sets of leaves in "0,1,2" Motzkin rooted trees with n edges. "0,1,2" trees are rooted trees where each vertex has out-degree zero, one, or two. They are counted by the Motzkin numbers A001006.
For "0,1,2" trees, Salaam (2008) proved that the g.f. of the number of r-sets of leaves is A000108(r-1) * z^(2*r-2) * T(z)^(2*r-1), where T(z) = 1/sqrt(1 - 2*z - 3*z^2) is the g.f. of the central trinomial numbers A002426.
For r = 2, we get a shifted version of A102839. For r = 3, we get twice of a shifted version of the current sequence. (End)

			From _Petros Hadjicostas_, Jun 03 2020: (Start)
Out of the A001006(4) = 9 Motzkin trees with n = 4 edges, only the following 2*a(4-4) = 2 have 3-sets of leaves:
            A                    A
           / \                  / \
          /   \                /   \
         B     C              B     C
        / \                        / \
       /   \                      /   \
      D     E                    D     E
      {C, D, E}                {B, D, E}
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 6.
Lifoma Salaam, Combinatorial statistics on phylogenetic trees, Ph.D. Dissertation, Howard University, Washington D.C., 2008; see Theorem 39 (p. 25).
J. Y. X. Yang, M. X. X. Zhong, and R. D. P. Zhou, On the Enumeration of (s, s+ 1, s+2)-Core Partitions, arXiv preprint arXiv:1406.2583 [math.CO], 2014. See Theorem 4.2.

Cf. A000108, A001006, A002426, A102839, A374506.

A[0]:= 1: A[1]:= 5:
for n from 2 to 100 do
A[n]:= (2+3/n)*A[n-1] + (3+9/n)*A[n-2]
od:
seq(A[n],n=0..100); # Robert Israel, Aug 01 2014

CoefficientList[Series[1/(1 - 2 x - 3 x^2)^(5/2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 01 2014 *)

x='x+O('x^50); Vec(1/(1-2*x-3*x^2)^(5/2)) \\ G. C. Greubel, Apr 06 2017

a(n) ~ 3^(n+3/2) * n^(3/2) / (8*sqrt(Pi)). - Vaclav Kotesovec, Jul 31 2014
a(n) = (2+3/n)*a(n-1) + (3+9/n)*a(n-2) for n >= 2. - Robert Israel, Aug 01 2014
a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} binomial(n+2,n-2*k) * binomial(2*k+2,k). - Seiichi Manyama, Jul 10 2024
a(n) = Sum_{k=0..n} (-2)^k * (3/2)^(n-k) * binomial(-5/2,k) * binomial(k,n-k). - Seiichi Manyama, Aug 23 2025

A375253 Expansion of (1 - 2x + 2x^2)/(1 - 2x - 3x^2)^(7/2).

Original entry on oeis.org

1, 5, 30, 140, 630, 2646, 10710, 41910, 159885, 597025, 2190188, 7914270, 28230020, 99567300, 347720040, 1203777072, 4135047615, 14105322315, 47813634330, 161154659820, 540353553894, 1803226621350, 5991410183850, 19827295283250, 65371101643575

Offset: 0

Seiichi Manyama, Aug 07 2024

nonn

Column k=4 of A091869 (with a different offset).
Cf. A374506, A375248.
Cf. A005717, A373651.

a[n_]:=(1+n)(2+n)(3+n)(4+n)Hypergeometric2F1[(1-n)/2,-n/2,2,4]/24; Array[a,25,0] (* Stefano Spezia, Aug 07 2024 *)

my(N=30, x='x+O('x^N)); Vec((1-2*x+2*x^2)/(1-2*x-3*x^2)^(7/2))

a(n) = (binomial(n+4,3)/4) * Sum_{k=0..floor(n/2)} binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = (binomial(n+4,3)/4) * A005717(n+1).
a(n) = ((n+4)/(n*(n+2))) * ((2*n+1)*a(n-1) + 3*(n+3)*a(n-2)).
a(n) = (1 + n)*(2 + n)*(3 + n)*(4 + n)*hypergeom([(1-n)/2, -n/2], [2], 4)/24. - Stefano Spezia, Aug 07 2024

A375260 Expansion of (1 - 3x + 9x^2 - 7x^3)/(1 - 2x - 3*x^2)^(7/2).

Original entry on oeis.org

1, 4, 30, 140, 665, 2856, 11844, 47160, 182655, 690580, 2560558, 9337692, 33573995, 119246960, 419034360, 1458687312, 5035531563, 17253821340, 58723235970, 198655153620, 668338862499, 2237229875496, 7454611712100, 24734393119800, 81748883914425

Offset: 0

Seiichi Manyama, Aug 08 2024

nonn

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

Cf. A374506, A375248, A375253.

Cf. A002426, A132885, A375259.

my(N=30, x='x+O('x^N)); Vec((1-3*x+9*x^2-7*x^3)/(1-2*x-3*x^2)^(7/2))

a(n) = binomial(n+3,3) * Sum_{k=0..floor(n/2)} binomial(n,n-2*k) * binomial(2*k,k).
a(n) = binomial(n+3,3) * A002426(n).
a(n) = A132885(n+6,3).
a(n) = ((n+3)/n^2) * ((2*n-1)*a(n-1) + 3*(n+2)*a(n-2)).

A374509 Expansion of 1/(1 - 2x + 5x^2)^(7/2).

Original entry on oeis.org

1, 7, 14, -42, -294, -462, 1386, 7722, 9009, -37037, -160160, -123760, 835380, 2848860, 1046520, -16550520, -45140865, 3533145, 296447690, 648593330, -393463070, -4895709390, -8489647530, 10975099590, 75528298755, 100311659721, -230350834728, -1097798696456

Offset: 0

Seiichi Manyama, Jul 09 2024

Cf. A098331, A102840, A374508.

Cf. A374506.

a[n_]:= Pochhammer[n+1, 6]*Hypergeometric2F1[(1-n)/2, -n/2, 4, -4]/6!; Array[a,28,0] (* Stefano Spezia, Jul 10 2024 *)

a(n) = binomial(n+6, 3)/20*sum(k=0, n\2, (-1)^k*binomial(n+3, n-2*k)*binomial(2*k+3, k));

a(0) = 1, a(1) = 7; a(n) = ((2*n+5)*a(n-1) - 5*(n+5)*a(n-2))/n.
a(n) = (binomial(n+6,3)/20) * Sum_{k=0..floor(n/2)} (-1)^k * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = Pochhammer(n+1, 6)*hypergeom([(1-n)/2, -n/2], [4], -4)/6!. - Stefano Spezia, Jul 10 2024
a(n) = (-1)^n * Sum_{k=0..n} 2^k * (5/2)^(n-k) * binomial(-7/2,k) * binomial(k,n-k). - Seiichi Manyama, Aug 23 2025

A375248 Expansion of (1 - x)/(1 - 2x - 3x^2)^(7/2).

Original entry on oeis.org

1, 6, 35, 168, 756, 3192, 12936, 50688, 193479, 722722, 2651649, 9581936, 34176324, 120526056, 420852204, 1456709328, 5002984791, 17062825626, 57827993685, 194871361608, 653285629920, 2179701604080, 7241015510820, 23958512912880, 78978801164445

Offset: 0

Seiichi Manyama, Aug 07 2024

nonn

First differences of A374506.
Cf. A046717, A109188, A371408.
Cf. A014531, A375253.

a[n_]:=(1+n)(2+n)(3+n)(4+n)(5+n)Hypergeometric2F1[(1-n)/2,-n/2,3,4]/120; Array[a,25,0] (* Stefano Spezia, Aug 07 2024 *)

my(N=30, x='x+O('x^N)); Vec((1-x)/(1-2*x-3*x^2)^(7/2))

a(n) = (binomial(n+5,3)/10) * Sum_{k=0..floor(n/2)} binomial(n+2,n-2*k) * binomial(2*k+2,k).
a(n) = (binomial(n+5,3)/10) * A014531(n+1).
a(n) = ((n+5)/(n*(n+4))) * ((2*n+3)*a(n-1) + 3*(n+4)*a(n-2)).
a(n) = (1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*hypergeom([(1-n)/2, -n/2], [3], 4)/120. - Stefano Spezia, Aug 07 2024

cp's OEIS Frontend

A245551 Expansion of 1/(1 - 2*x - 3*x^2)^(5/2).

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A375253 Expansion of (1 - 2*x + 2*x^2)/(1 - 2*x - 3*x^2)^(7/2).

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A375260 Expansion of (1 - 3*x + 9*x^2 - 7*x^3)/(1 - 2*x - 3*x^2)^(7/2).

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A374509 Expansion of 1/(1 - 2*x + 5*x^2)^(7/2).

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A375248 Expansion of (1 - x)/(1 - 2*x - 3*x^2)^(7/2).

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A245551 Expansion of 1/(1 - 2x - 3x^2)^(5/2).

A375253 Expansion of (1 - 2x + 2x^2)/(1 - 2x - 3x^2)^(7/2).

A375260 Expansion of (1 - 3x + 9x^2 - 7x^3)/(1 - 2x - 3*x^2)^(7/2).

A374509 Expansion of 1/(1 - 2x + 5x^2)^(7/2).

A375248 Expansion of (1 - x)/(1 - 2x - 3x^2)^(7/2).