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A187661 Binomial convolution of the (signless) central Stirling numbers of the first kind and the central Stirling numbers of the second kind.

%I A187661 #17 May 28 2017 04:07:52
%S A187661 1,2,20,369,10192,379850,17930697,1027046517,69216504576,
%T A187661 5363945384274,469658243947850,45827641349686636,4928867833029014503,
%U A187661 579101340954599901152,73778702335232336908585,10129059530832922239925140
%N A187661 Binomial convolution of the (signless) central Stirling numbers of the first kind and the central Stirling numbers of the second kind.
%H A187661 G. C. Greubel, <a href="/A187661/b187661.txt">Table of n, a(n) for n = 0..250</a>
%F A187661 a(n) = Sum_{k=0..n} binomial(n,k) * s(2*k,k) * S(2*n-2*k,n-k).
%F A187661 a(n) ~ m * n^n * c^(2*n) * 2^(3*n-1) / (sqrt(Pi*(c-1)*n) * exp(n) * (2*c-1)^n), where c = -LambertW(-1,-exp(-1/2)/2) = 1.75643120862616967698..., and m = Sum_{j>=0} StirlingS2(2*j,j) * (2*c-1)^j / (j! * 2^(3*j) * c^(2*j)) = 1.170003674502655133465266152119563086693466... . - _Vaclav Kotesovec_, May 22 2014
%p A187661 seq(sum(binomial(n,k) * abs(combinat[stirling1](2*k, k)) * combinat[stirling2](2*(n-k), n-k), k=0..n), n=0..12);
%t A187661 Table[Sum[Binomial[n, k]Abs[StirlingS1[2k, k]]StirlingS2[2n - 2k, n - k], {k, 0, n}], {n, 0, 15}]
%o A187661 (Maxima) makelist(sum(binomial(n,k)*abs(stirling1(2*k,k))*stirling2(2*n-2*k,n-k),k,0,n),n,0,12);
%o A187661 (PARI) a(n) = sum(k=0, n, binomial(n,k)*abs(stirling(2*k, k, 1)*stirling(2*(n-k), n-k, 2))); \\ _Michel Marcus_, May 28 2017
%Y A187661 Cf. A007820, A187646.
%K A187661 nonn,easy
%O A187661 0,2
%A A187661 _Emanuele Munarini_, Mar 12 2011