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Eigensequence of triangle A085478: a(n) = Sum_{k=0..n-1} A085478(n-1,k)*a(k) for n > 0 with a(0) = 1.
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%I #31 May 31 2022 11:21:25

%S 1,1,2,6,23,106,567,3434,23137,171174,1376525,11934581,110817423,

%T 1095896195,11487974708,127137087319,1480232557526,18075052037054,

%U 230855220112093,3076513227516437,42686898298650967,615457369662333260

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

%H Seiichi Manyama, <a href="/A125273/b125273.txt">Table of n, a(n) for n = 0..517</a>

%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>. [Cached copy]

%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.

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

%F G.f. satisfies: A(x) = 1 + x*A(x/(1-x)^2) / (1-x). - _Paul D. Hanna_, Aug 15 2007

%e a(3) = 1*(1) + 3*(1) + 1*(2) = 6;

%e a(4) = 1*(1) + 6*(1) + 5*(2) + 1*(6) = 23;

%e a(5) = 1*(1) + 10*(1) + 15*(2) + 7*(6) + 1*(23) = 106.

%e Triangle A085478(n,k) = binomial(n+k, n-k) (with rows n >= 0 and columns k = 0..n) begins:

%e 1;

%e 1, 1;

%e 1, 3, 1;

%e 1, 6, 5, 1;

%e 1, 10, 15, 7, 1;

%e 1, 15, 35, 28, 9, 1;

%e ...

%e where g.f. of column k = 1/(1-x)^(2*k+1).

%t 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 *)

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

%Y Cf. A085478, A125274 (variant), A351813.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 26 2006