A125274 Eigensequence of triangle A078812: a(n) = Sum_{k=0..n-1} A078812(n-1,k)*a(k) for n > 0 with a(0)=1.

1, 1, 3, 10, 42, 210, 1199, 7670, 54224, 418744, 3499781, 31425207, 301324035, 3069644790, 33078375153, 375634524357, 4480492554993, 55971845014528, 730438139266281, 9935106417137098, 140553930403702487

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..516

Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2.

Formula

a(n) = Sum_{k=0..n-1} binomial(n+k, n-k-1)*a(k) for n > 0 with a(0)=1.

G.f. satisfies A(x) = 1 + x/(1-x)^2*A(x/(1-x)^2). [Vladimir Kruchinin, Nov 28 2011]

Example

a(3) = 3*(1) + 4*(1) + 1*(3) = 10;
a(4) = 4*(1) + 10*(1) + 6*(3) + 1*(10) = 42;
a(5) = 5*(1) + 20*(1) + 21*(3) + 8*(10) + 1*(42) = 210.
Triangle A078812(n,k) = binomial(n+k+1, n-k) begins:
  1;
  2,  1;
  3,  4,  1;
  4, 10,  6,  1;
  5, 20, 21,  8,  1;
  6, 35, 56, 36, 10,  1; ...
where g.f. of column k = 1/(1-x)^(2*k+2).

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n+k, n-k-1] * a[k], {k, 0, n-1}]; Array[a, 20, 0] (* Amiram Eldar, Nov 24 2018 *)

Prog

(PARI) a(n)=if(n==0, 1, sum(k=0, n-1, a(k)*binomial(n+k, n-k-1)))

Crossrefs

Cf. A078812, A125273 (variant), A351816, A351817, A351818.

Sequence in context: A300632 A300511 A007552 * A030903 A030816 A030964

Adjacent sequences: A125271 A125272 A125273 * A125275 A125276 A125277

Keyword

nonn

Author

Paul D. Hanna, Nov 26 2006

Status

approved