A124890 -id:A124890 - cp's OEIS frontend

A124891 L.g.f.: A(x) = log(G124890(x)) where G124890(x) is the g.f. of A124890.

1, 5, 25, 137, 766, 4379, 25355, 148273, 873574, 5177450, 30833342, 184355207, 1105977887, 6653964847, 40131748300, 242567280865, 1468928473132, 8910461020730, 54131814523902, 329299899410062, 2005674943792559

Offset: 0

Paul D. Hanna, Nov 12 2006

nonn

			A(x) = x + 5*x^2/2 + 25*x^3/3 + 137*x^4/4 + 766*x^5/5 + 4379*x^6/6 +...
exp(A(x)) = G124890(x) where
G124890(x) = 1 + x + 3*x^2 + 11*x^3 + 47*x^4 + 216*x^5 + 1047*x^6 +...

Cf. A124890, A124328, A124889.

a(n) = A124328(2n+2,n+1) for n>=0; thus a(n) is the number of ordered trees with 2(n+1) edges, with thinning limbs and with root of degree n+1.

A124328 Triangle read by rows: T(n,k) is the number of ordered trees with n edges, with thinning limbs and with root of degree k (1<=k<=n). An ordered tree with thinning limbs is such that if a node has k children, all its children have at most k children.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 10, 9, 4, 1, 1, 22, 25, 14, 5, 1, 1, 46, 69, 44, 20, 6, 1, 1, 101, 186, 137, 70, 27, 7, 1, 1, 220, 503, 416, 235, 104, 35, 8, 1, 1, 492, 1356, 1256, 766, 375, 147, 44, 9, 1, 1, 1104, 3663, 3760, 2465, 1296, 567, 200, 54, 10, 1, 1, 2515, 9907

Offset: 1

Emeric Deutsch, Nov 03 2006

Row sums yield A124344. T(n,2) = A124329(n).

			Triangle starts:
  1;
  1,1;
  1,2,1;
  1,5,3,1;
  1,10,9,4,1;

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A124344, A124329.

Cf. A124889, A124890, A124891.

t[n_, n_] = 1; t[n_, k_] /; n == k + 1 := t[n, k] = n - 1; t[n_, k_] := t[n, k] = Coefficient[(1 + x*Sum[ x^(r - 1)*Sum[ t[r, c], {c, 1, k }], {r, 1, n - k}] + x^n)^k, x, n - k ]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 21 2013, after Paul D. Hanna *)

{T(n,k)=if(n==k,1,if(n==k+1,n-1,polcoeff( (1 + x*sum(r=1,n-k,x^(r-1)*sum(c=1,k,T(r,c)))+x*O(x^n))^k,n-k)))} \\ Paul D. Hanna, Nov 12 2006

The g.f. F[k]=F[k](z) of column k satisfies F[k]=(F[k-1]^(1/(k-1)) + z*F[k])^k; F[1]=z/(1-z).

Central terms are: T(2*n-1,n) = A124889(n-1), T(2*n,n) = A124891(n-1), for n>=1. - Paul D. Hanna, Nov 12 2006

More terms from Paul D. Hanna, Nov 12 2006

A124889 Central terms of triangle A124328; a(n) = A124328(2n+1,n+1) for n>=0.

Original entry on oeis.org

1, 2, 9, 44, 235, 1296, 7329, 42104, 244719, 1434840, 8470517, 50280372, 299806572, 1794382548, 10773855210, 64865425760, 391457133672, 2367314649522, 14342441005900, 87035702613500, 528938145043707, 3218713327609648

Offset: 0

Paul D. Hanna, Nov 12 2006

nonn

Cf. A124328, A124889, A124890, A124891.

a(n) = (n+1)*A124890(n).

cp's OEIS Frontend

A124891 L.g.f.: A(x) = log(G124890(x)) where G124890(x) is the g.f. of A124890.

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A124328 Triangle read by rows: T(n,k) is the number of ordered trees with n edges, with thinning limbs and with root of degree k (1<=k<=n). An ordered tree with thinning limbs is such that if a node has k children, all its children have at most k children.

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A124889 Central terms of triangle A124328; a(n) = A124328(2n+1,n+1) for n>=0.

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