A298753 Triangle read by rows, T(n,m) = Sum_{k=1..m} k*k!*(-1)^(m+k)*Stirling2(m,k)* C(2*n+k-2*m-1,n-m)/(n+k-m), for n >= 0 and 0 <= m <= n.

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 5, 8, 7, 1, 0, 14, 23, 26, 15, 1, 0, 42, 70, 89, 80, 31, 1, 0, 132, 222, 302, 335, 242, 63, 1, 0, 429, 726, 1032, 1294, 1265, 728, 127, 1, 0, 1430, 2431, 3564, 4842, 5654, 4823, 2186, 255, 1, 0, 4862, 8294, 12441, 17886, 23472, 25270, 18569, 6560, 511, 1, 0, 16796, 28730, 43862, 65767, 93732

Offset

0,8

Links

Table of n, a(n) for n=0..71.

Formula

E.g.f.: 1/(1+C(x)*(exp(-x*y)-1)), where C(x)=A000108(x) is the g.f. of Catalan numbers.

T(n,m) = Sum_{k=1..m} k*k!*(-1)^(m+k)*Stirling2(m,k)*C(2*n+k-2*m-1,n-m)/(n+k-m), m<n, T(n,n)=1.

T(n,m) = Sum_{k=0..n-m} (m+k)*E(m,k)*C(2*n-m-k-1,n-m-k)/n, n>m, T(n,n)=1, E(n,m) is Euler triangle A008292.

Example

Triangle begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   3,    1;
  0,   5,   8,    7,    1;
  0,  14,  23,   26,   15,    1;
  0,  42,  70,   89,   80,   31,   1;
  0, 132, 222,  302,  335,  242,  63,   1;
  0, 429, 726, 1032, 1294, 1265, 728, 127, 1;

Maple

T := (n, m) -> `if`(n=0, 1,
add(combinat[eulerian1](m, k-m)*binomial(2*n-k-1, n-k)*k/n, k=m..m+min(m, n-m))):
for n from 0 to 8 do seq(T(n, k), k=0..n) od; # Peter Luschny, Jan 26 2018

Prog

(Maxima)
T(n, m):=if n<m then 0 else if n=m then 1 else sum(k*k!*(-1)^(m+k)*stirling2(m, k)* binomial(2*n+k-2*m-1, n-m)/(n+k-m), k, 1, m);

Crossrefs

Cf. A000108, A008292.

Sequence in context: A332645 A344855 A081247 * A173050 A172380 A144633

Adjacent sequences: A298750 A298751 A298752 * A298754 A298755 A298756

Keyword

nonn,tabl

Author

Vladimir Kruchinin, Jan 26 2018

Status

approved