A188109 Triangle T(n,m), [x*A(x)]^m=sum(n>=m T(n,m)*x^n), where A(x) satisfies x*A(x)^2= -(2*x*A(x)+sqrt(1-4*x*A(x))-1)/(4*x*A(x)+sqrt(1-4*x*A(x))-1)

1, 3, 1, 19, 6, 1, 152, 47, 9, 1, 1367, 418, 84, 12, 1, 13195, 4007, 825, 130, 15, 1, 133556, 40368, 8433, 1400, 185, 18, 1, 1398696, 421332, 88872, 15239, 2170, 249, 21, 1, 15029311, 4515706, 959080, 168112, 25100, 3162, 322, 24, 1, 164764985, 49405895, 10547361, 1878462, 289788, 38772, 4403, 404, 27, 1

Offset

1,2

Links

Table of n, a(n) for n=1..55.

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

T(n,m) = m/n*A035324(2*n-m,n).

Example

1;
3, 1;
19, 6, 1;
152, 47, 9, 1;
1367, 418, 84, 12, 1;
13195, 4007, 825, 130, 15, 1;
133556, 40368, 8433, 1400, 185, 18, 1;
1398696, 421332, 88872, 15239, 2170, 249, 21, 1

Mathematica

(* S = A035324 *)
S[n_, m_] /; n >= m >= 1 := S[n, m] = 2(2(n-1)+m)(S[n-1, m]/n) + m(S[n-1, m-1]/n); S[n_, m_] /; n < m = 0; S[n_, 0] = 0; S[1, 1] = 1;
T[n_, m_] := m/n S[2n-m, n];
Table[T[n, m], {n, 1, 10}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 16 2019 *)

Crossrefs

Cf. A035324

Sequence in context: A016480 A086156 A227888 * A247232 A147076 A372175

Adjacent sequences: A188106 A188107 A188108 * A188110 A188111 A188112

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Mar 21 2011

Status

approved