A126190 -id:A126190 - cp's OEIS frontend

A387239 a(n) = Sum_{k=0..n} binomial(n+3,k+3) * binomial(2*k+6,k+6).

1, 12, 95, 630, 3801, 21672, 119154, 639180, 3369795, 17543196, 90476100, 463291920, 2359240975, 11961944400, 60440659640, 304543085040, 1531044995355, 7682898791700, 38494752520175, 192632866196694, 962948703201331, 4809438625979592, 24002988378037350, 119719958370912900

Offset: 0

Seiichi Manyama, Aug 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A026376, A026377.

Cf. A126190.

[&+[Binomial(n+3,k+3) * Binomial(2*k+6,k+6): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 24 2025

Table[Sum[Binomial[n+3,k+3]* Binomial[2*k+6, k+6],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 24 2025 *)

a(n) = sum(k=0, n, binomial(n+3, k+3)*binomial(2*k+6, k+6));

n*(n+6)*a(n) = (n+3) * (3*(2*n+5)*a(n-1) - 5*(n+2)*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 3^(n-2*k) * binomial(n+3,n-2*k) * binomial(2*k+3,k).
a(n) = [x^n] (1+3*x+x^2)^(n+3).
E.g.f.: exp(3*x) * BesselI(3, 2*x), with offset 3.

A387238 Expansion of 1/((1-x) * (1-5*x))^(7/2).

Original entry on oeis.org

1, 21, 266, 2646, 22806, 178794, 1310694, 9140274, 61330269, 399107709, 2533330800, 15751925280, 96257031780, 579556206180, 3445117599480, 20252115155160, 117890464642335, 680320688005035, 3895668955041710, 22152779612619810, 125183331416173030

Offset: 0

Seiichi Manyama, Aug 23 2025

Vincenzo Librandi, Table of n, a(n) for n = 0..500

Cf. A026375, A385563, A387237.

Cf. A126190, A374509, A387239.

R := PowerSeriesRing(Rationals(), 34); f := 1/((1-x) * (1-5*x))^(7/2); coeffs := [ Coefficient(f, n) : n in [0..33] ]; coeffs; // Vincenzo Librandi, Aug 24 2025

CoefficientList[Series[1/((1-x)*(1-5*x))^(7/2),{x,0,33}],x] (* Vincenzo Librandi, Aug 24 2025 *)

my(N=30, x='x+O('x^N)); Vec(1/((1-x)*(1-5*x))^(7/2))

n*a(n) = (6*n+15)*a(n-1) - 5*(n+5)*a(n-2) for n > 1.
a(n) = (-1)^n * Sum_{k=0..n} 5^k * binomial(-7/2,k) * binomial(-7/2,n-k).
a(n) = Sum_{k=0..n} (-4)^k * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = Sum_{k=0..n} 4^k * 5^(n-k) * binomial(-7/2,k) * binomial(n+6,n-k).
a(n) = (binomial(n+6,3)/20) * A387239(n).
a(n) = (-1)^n * Sum_{k=0..n} 6^k * (5/6)^(n-k) * binomial(-7/2,k) * binomial(k,n-k).

A126188 Triangle read by rows: T(n,k) is the number of hex trees with n edges and k pairs of adjacent vertices of outdegree 2.

Original entry on oeis.org

1, 3, 10, 36, 135, 2, 519, 24, 2034, 180, 5, 8100, 1110, 75, 32688, 6210, 675, 14, 133380, 32886, 4851, 252, 549342, 168210, 30996, 2646, 42, 2280690, 840132, 184842, 21672, 882, 9534591, 4124682, 1053486, 154980, 10584, 132, 40103019

Offset: 0

Emeric Deutsch, Dec 25 2006

A hex tree is a rooted tree where each vertex has 0, 1, or 2 children and, when only one child is present, it is either a left child, or a middle child, or a right child (name due to an obvious bijection with certain tree-like polyhexes; see the Harary-Read reference).
Row n has floor(n/2) terms (n>=2).
Sum of terms in row n = A002212(n+1).
T(n,0) = A126189(n).
Sum_{k=0..floor(n/2)-1} k*T(n,k) = A126190(n).

			Triangle starts:
     1;
     3;
    10;
    36;
   135,    2;
   519,   24;
  2034,  180,    5;
  8100, 1110,   75;

F. Harary and R. C. Read, The enumeration of tree-like polyhexes, Proc. Edinburgh Math. Soc. (2) 17 (1970), 1-13.

Cf. A002212, A126189, A126190.

G:=1/2*(12*z^3*t+2*z^2*t^2-2*z^2*t-6*z^3*t^2-3*z-6*z^3+1-sqrt(1+9*z^2-4*z^2*t-6*z+12*z^3*t-12*z^3))/z^2/(3*z*t-t-3*z)^2: Gser:=simplify(series(G,z=0,18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser,z,n)) od: 1;3;for n from 2 to 14 do seq(coeff(P[n],t,j),j=0..floor(n/2)-1) od;

g[t_,z_] = G /. Solve[G == 1 + 3z*G + z^2*(1 + 3z*G + t*(G - 1 - 3z*G))^2, G][[1]]; Flatten[ CoefficientList[ CoefficientList[ Series[g[t,z], {z,0,13}], z], t]][[1 ;; 39]] (* Jean-François Alcover, May 27 2011, after g.f. *)

G.f.: G = G(t,z) = 1+3*z*G+z^2*(1+3*z*G+t*(G-1-3*z*G))^2 (explicit expression in the Maple program).

cp's OEIS Frontend

A387239 a(n) = Sum_{k=0..n} binomial(n+3,k+3) * binomial(2*k+6,k+6).

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A387238 Expansion of 1/((1-x) * (1-5*x))^(7/2).

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A126188 Triangle read by rows: T(n,k) is the number of hex trees with n edges and k pairs of adjacent vertices of outdegree 2.

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Maple

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