Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #30 May 04 2024 06:34:13
%S 1,1,1,1,0,1,-5,29,-196,1518,-13266,129163,-1386572,16270671,
%T -207195495,2845705719,-41930575740,659781404944,-11041824881696,
%U 195839234324062,-3669384701403344,72423881548363354,-1501924519315744146,32649768696532126439,-742432111781693213350
%N Stirling1 transform of Catalan numbers: Sum_{k=0..n} Stirling1(n,k)*binomial(2*k,k)/(k+1).
%C 1, 1, 1, 0, 1, -5, 29, -196, ... is the Stirling1 transform of the Motzkin numbers A001006. - _Philippe Deléham_, May 27 2015
%H Andrew Howroyd, <a href="/A086672/b086672.txt">Table of n, a(n) for n = 0..200</a>
%F E.g.f.: hypergeom([1/2], [2], 4*log(1+x)) = (1+x)^2*(BesselI(0, 2*log(1+x))-BesselI(1, 2*log(1+x))).
%F Let C(m) be the m-th Catalan number, A000108(m). Let S(m, n) = an unsigned Stirling number of the first kind. Then a(m) = sum{k=0 to m} S(m, k) C(k) (-1)^(k+m). - _Leroy Quet_, Jan 23 2004
%F E.g.f. f(x) satisfies f(x) = 1 + integral{0 to x} f(y) f((x-y)/(1+y))/(1+y) dy. - _Leroy Quet_, Jan 25 2004
%F a(n) = Sum_{k = 0..n} A048994(n, k) * A000108(k). - _Philippe Deléham_, May 27 2015
%F a(n+1) = Sum_{k = 0..n} A048994(n,k) * A001006(k). - _Philippe Deléham_, May 27 2015
%F For n > 1, a(n) = (A201950(n+1) - (3*n-2)*A201950(n) + n*(3*n-7)*A201950(n-1) - (n-4)*(n-1)*n*A201950(n-2)) * (-1)^n/2. - _Vaclav Kotesovec_, May 04 2024
%t Table[Sum[StirlingS1[n, k] * CatalanNumber[k], {k, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Aug 04 2021 *)
%o (PARI) a(n)={sum(k=0, n, stirling(n,k,1) * binomial(2*k, k) / (k+1))} \\ _Andrew Howroyd_, Jan 27 2020
%Y Cf. A000108, A001006, A008275, A048994, A064856.
%Y Cf. A201950, A201952, A306335.
%K easy,sign
%O 0,7
%A _Vladeta Jovovic_, Sep 12 2003
%E Terms a(21) and beyond from _Andrew Howroyd_, Jan 27 2020