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
Keywords
Examples
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).
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.
Programs
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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 *) -
PARI
a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(n+k, n-k-1)))
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]