OFFSET
1,5
COMMENTS
Equals binomial transform of an infinite lower triangular matrix with A001006 as the main diagonal and the rest zeros. [Gary W. Adamson, Dec 31 2008] [Corrected by Paul Barry, Mar 06 2011]
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows 1..150, flattened)
J.-L. Baril, S. Kirgizov, The pure descent statistic on permutations, Preprint, 2016.
J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combinat. Theory, Ser A, 19, 214-222, 1975.
Lin Yang and Shengliang Yang, Protected Branches in Ordered Trees, J. Math. Study (2023) Vol. 56, No. 1, 1-17.
FORMULA
T(n,k) = M(k-1)*binomial(n-1, k-1), where M(k) = A001006(k) = (Sum_{q=0..ceiling((k+1)/2)} binomial(k+1, q)*binomial(k+1-q, q-1))/(k+1) is a Motzkin number.
G.f.: G = G(t,z) satisfies t*z*G^2 -(1 - z + t*z)*G + 1- z + t*z = 0.
From Paul Barry, Mar 06 2011: (Start)
G.f.: 1/(1-x-xy-x^2y^2/(1-x-xy-x^2y^2/(1-x-xy-x^2y^2/(1-... (continued fraction).
G.f.: (1-x(1+y)-sqrt(1-2x(1+y)+x^2(1+2y-3y^2)))/(2x^2*y^2).
E.g.f.: exp(x(1+y))*Bessel_I(1,2*x*y)/(x*y). (End)
EXAMPLE
Triangle begins:
1;
1, 1;
1, 2, 2;
1, 3, 6, 4;
1, 4, 12, 16, 9;
1, 5, 20, 40, 45, 21;
1, 6, 30, 80, 135, 126, 51;
1, 7, 42, 140, 315, 441, 357, 127;
MAPLE
M := n->sum(binomial(n+1, q)*binomial(n+1-q, q-1), q=0..ceil((n+1)/2))/(n+1): T := (n, k)->binomial(n-1, k-1)*M(k-1): seq(seq(T(n, k), k=1..n), n=1..13);
MATHEMATICA
(* m = MotzkinNumber *) m[0] = 1; m[n_] := m[n] = m[n - 1] + Sum[m[k]*m[n - 2 - k], {k, 0, n - 2}]; t[n_, k_] := m[k - 1]*Binomial[n - 1, k - 1]; Table[t[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 10 2013 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Emeric Deutsch, Feb 23 2004
STATUS
approved