A125273 Eigensequence of triangle A085478: a(n) = Sum_{k=0..n-1} A085478(n-1,k)*a(k) for n > 0 with a(0) = 1.

1, 1, 2, 6, 23, 106, 567, 3434, 23137, 171174, 1376525, 11934581, 110817423, 1095896195, 11487974708, 127137087319, 1480232557526, 18075052037054, 230855220112093, 3076513227516437, 42686898298650967, 615457369662333260

Offset: 0

Author

Paul D. Hanna, Nov 26 2006

Keywords

• nonn

Examples

			a(3) = 1*(1) + 3*(1) + 1*(2) = 6;
a(4) = 1*(1) + 6*(1) + 5*(2) + 1*(6) = 23;
a(5) = 1*(1) + 10*(1) + 15*(2) + 7*(6) + 1*(23) = 106.
Triangle A085478(n,k) = binomial(n+k, n-k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  5,  1;
  1, 10, 15,  7,  1;
  1, 15, 35, 28,  9,  1;
  ...
where g.f. of column k = 1/(1-x)^(2*k+1).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..517
• Guo-Niu Han, Enumeration of Standard Puzzles. [Cached copy]
• Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
• Jeffrey B. Remmel, Consecutive Up-down Patterns in Up-down Permutations, Electron. J. Combin., 21 (2014), #P3.2.

Crossrefs

Cf. A085478, A125274 (variant), A351813.

Programs

Mathematica

A125273=ConstantArray[0,20]; A125273[[1]]=1; Do[A125273[[n]]=1+Sum[A125273[[k]]*Binomial[n+k-1, n-k-1],{k,1,n-1}];,{n,2,20}]; Flatten[{1,A125273}] (* Vaclav Kotesovec, Dec 10 2013 *)

PARI

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

Formula

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

G.f. satisfies: A(x) = 1 + x*A(x/(1-x)^2) / (1-x). - Paul D. Hanna, Aug 15 2007