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A051163 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a1,a1,a2,a2,a3,a3,...).

%I A051163 #36 Apr 10 2022 09:47:44
%S A051163 1,2,5,12,30,76,194,496,1269,3250,8337,21428,55184,142376,367916,
%T A051163 952000,2466014,6393372,16586678,43054344,111801908,290412296,
%U A051163 754543052,1960808160,5096293794,13247503540,34440553562,89549255592,232868582328,605646682144
%N A051163 Sequence is defined by property that (a0,a1,a2,a3,...) = binomial transform of (a0,a0,a1,a1,a2,a2,a3,a3,...).
%C A051163 Equals the self-convolution of A027826. Also equals antidiagonal sums of symmetric square array A100936. - _Paul D. Hanna_, Nov 22 2004
%C A051163 Equals eigensequence of triangle A152198. - _Gary W. Adamson_, Nov 28 2008
%H A051163 Alois P. Heinz, <a href="/A051163/b051163.txt">Table of n, a(n) for n = 0..1000</a>
%H A051163 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A051163 a(n) = 1 + Sum_{k=1..n} Sum_{j=0..n-k} C(k, j)*C(n-k, j)*a(j). - _Paul D. Hanna_, Nov 22 2004
%F A051163 G.f. A(x) satisfies: A(x) = A(x^2/(1-x)^2)/(1-x)^2 and A(x^2) = A(x/(1+x))/(1+x)^2. - _Paul D. Hanna_, Nov 22 2004
%F A051163 a(0) = 1; a(n) = Sum_{k=0..floor(n/2)} binomial(n+1,2*k+1) * a(k). - _Ilya Gutkovskiy_, Apr 07 2022
%p A051163 a:= proc(n) option remember; add(`if`(k<2, 1,
%p A051163       a(iquo(k, 2)))*binomial(n, k), k=0..n)
%p A051163     end:
%p A051163 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 08 2015
%t A051163 a[n_] := a[n] = 1 + Sum[Binomial[k, j]*Binomial[n-k, j]*a[j], {k, 1, n}, {j, 0, n-k}]; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Nov 11 2015 *)
%o A051163 (PARI) a(n)=1+sum(k=1,n,sum(j=0,n-k,binomial(k,j)*binomial(n-k,j)*a(j)))
%o A051163 (PARI) a(n)=local(A,m); if(n<0,0,m=1; A=1+O(x); while(m<=n,m*=2; A=subst(A,x,(x/(1-x))^2)/(1-x)); polcoeff(A^2,n))
%o A051163 for(n=0,40,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 22 2004
%Y A051163 Cf. A051164, A051165, A051166, A027826, A100936, A100937, A152198.
%K A051163 easy,nonn,eigen
%O A051163 0,2
%A A051163 _Jonas Wallgren_
%E A051163 More terms from _Vladeta Jovovic_, Jul 26 2002