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A283298 Diagonal of the Euler-Seidel matrix for the Catalan numbers.

%I A283298 #56 Sep 01 2025 03:05:41
%S A283298 1,3,26,305,4120,60398,934064,15000903,247766620,4182015080,
%T A283298 71816825856,1250772245698,22039796891026,392213323252200,
%U A283298 7038863826811100,127248841020380105,2315130641074743540,42358284517663463380,778876539384226875800
%N A283298 Diagonal of the Euler-Seidel matrix for the Catalan numbers.
%H A283298 Paul Barry and A. Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry2/barry94r.html">The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences</a>, J. Int. Seq. 13 (2010) # 10.8.2, Example 11.
%F A283298 a(n) = Sum_{i=0..n} binomial(n,i) * A000108(n+i).
%F A283298 D-finite with recurrence 2*n*(2*n+1)*(9*n-11)*a(n) +(-711*n^3+1589*n^2-986*n+144)*a(n-1) -10*(n-1)*(9*n-2)*(2*n-3)*a(n-2)=0.
%F A283298 a(n) ~ 2^(2*n) * 5^(n + 3/2) / (27 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Sep 01 2025
%p A283298 A000108 := n-> binomial(2*n, n)/(n+1):
%p A283298 A283298 := proc(n)
%p A283298     add(binomial(n,i)*A000108(n+i),i=0..n) ;
%p A283298 end proc:
%p A283298 seq(A283298(n),n=0..30) ;
%t A283298 Table[Sum[Binomial[n, i] CatalanNumber[n + i], {i, 0, n}], {n, 0, 50}] (* _Indranil Ghosh_, Jul 20 2017 *)
%o A283298 (Python)
%o A283298 from sympy import binomial, catalan
%o A283298 def a(n): return sum(binomial(n, i)*catalan(n + i) for i in range(n + 1))
%o A283298 print([a(n) for n in range(51)]) # _Indranil Ghosh_, Jul 20 2017
%o A283298 (PARI) C(n) = binomial(2*n,n)/(n+1); \\ A000108
%o A283298 a(n) = sum(i=0, n, binomial(n,i) * C(n+i)); \\ _Michel Marcus_, Nov 12 2022
%Y A283298 Central elements of rows in A106534, A280470.
%Y A283298 Cf. A000108.
%K A283298 nonn,easy,changed
%O A283298 0,2
%A A283298 _R. J. Mathar_, Jul 20 2017