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A176675 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=0, k=0 and l=-1.

%I A176675 #27 Dec 19 2024 17:14:53
%S A176675 1,0,-1,-3,-7,-14,-23,-24,21,220,821,2261,4935,7814,3615,-34251,
%T A176675 -179511,-593420,-1521779,-3001089,-3500101,4410846,44902907,
%U A176675 179440150,526896835,1212055740,1976482795,697522595,-10858119895,-56563343774
%N A176675 Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=0, k=0 and l=-1.
%F A176675 G.f.: f(z) = (1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=0, l=-1).
%F A176675 a(n) = sum(k=0..n, (A000108(k) * sum(i=0..n-k, binomial(k+1,n-k-i)*binomial(k+i,k)*(-2)^(n-k-i)))), where A000108 is the Catalan numbers. [_Vladimir Kruchinin_, Nov 11 2012]
%F A176675 Conjecture: +(n+1)*a(n) +2*(-3*n+1)*a(n-1) +(13*n-21)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Feb 29 2016
%e A176675 a(2)=2*0*1-1=-1. a(2)=2*1*(-1)+0^2-1=-3. a(4)=2*1*(-3)+2*0*(-1)-1=-7.
%p A176675 l:=-1: : k := 0 : m:=0: d(0):=1:d(1):=m: for n from 1 to 30 do d(n+1):=sum(d(p)*d(n-p)+k,p=0..n)+l:od :
%p A176675 taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z),z=0,30);seq(d(n),n=0..30);
%o A176675 (Maxima) a(n):=sum((binomial(2*k, k)*sum(binomial(k+1, n-k-i)*binomial(k+i, k)*(-2)^(n-k-i), i, 0, n-k))/(k+1), k, 0, n); /* _Vladimir Kruchinin_, Nov 16 2012 */
%o A176675 (PARI) /* Using Vladimir Kruchinin's formula: */
%o A176675 {A000108(k)=binomial(2*k,k)/(k+1)}
%o A176675 {a(n)=sum(k=0,n,(A000108(k)*sum(i=0,n-k, binomial(k+1,n-k-i)*binomial(k+i,k)*(-2)^(n-k-i))))} \\ _Paul D. Hanna_, Nov 15 2012
%Y A176675 Cf. A176648.
%K A176675 easy,sign
%O A176675 0,4
%A A176675 _Richard Choulet_, Apr 23 2010