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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.

%I A125274 #22 May 31 2022 11:21:13
%S A125274 1,1,3,10,42,210,1199,7670,54224,418744,3499781,31425207,301324035,
%T A125274 3069644790,33078375153,375634524357,4480492554993,55971845014528,
%U A125274 730438139266281,9935106417137098,140553930403702487
%N 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.
%H A125274 Seiichi Manyama, <a href="/A125274/b125274.txt">Table of n, a(n) for n = 0..516</a>
%H A125274 Jeffrey B. Remmel, <a href="https://doi.org/10.37236/3210">Consecutive Up-down Patterns in Up-down Permutations</a>, Electron. J. Combin., 21 (2014), #P3.2.
%F A125274 a(n) = Sum_{k=0..n-1} binomial(n+k, n-k-1)*a(k) for n > 0 with a(0)=1.
%F A125274 G.f. satisfies A(x) = 1 + x/(1-x)^2*A(x/(1-x)^2). [_Vladimir Kruchinin_, Nov 28 2011]
%e A125274 a(3) = 3*(1) + 4*(1) + 1*(3) = 10;
%e A125274 a(4) = 4*(1) + 10*(1) + 6*(3) + 1*(10) = 42;
%e A125274 a(5) = 5*(1) + 20*(1) + 21*(3) + 8*(10) + 1*(42) = 210.
%e A125274 Triangle A078812(n,k) = binomial(n+k+1, n-k) begins:
%e A125274   1;
%e A125274   2,  1;
%e A125274   3,  4,  1;
%e A125274   4, 10,  6,  1;
%e A125274   5, 20, 21,  8,  1;
%e A125274   6, 35, 56, 36, 10,  1; ...
%e A125274 where g.f. of column k = 1/(1-x)^(2*k+2).
%t A125274 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 *)
%o A125274 (PARI) a(n)=if(n==0,1,sum(k=0,n-1, a(k)*binomial(n+k, n-k-1)))
%Y A125274 Cf. A078812, A125273 (variant), A351816, A351817, A351818.
%K A125274 nonn
%O A125274 0,3
%A A125274 _Paul D. Hanna_, Nov 26 2006