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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,...).
11
1, 2, 5, 12, 30, 76, 194, 496, 1269, 3250, 8337, 21428, 55184, 142376, 367916, 952000, 2466014, 6393372, 16586678, 43054344, 111801908, 290412296, 754543052, 1960808160, 5096293794, 13247503540, 34440553562, 89549255592, 232868582328, 605646682144
OFFSET
0,2
COMMENTS
Equals the self-convolution of A027826. Also equals antidiagonal sums of symmetric square array A100936. - Paul D. Hanna, Nov 22 2004
Equals eigensequence of triangle A152198. - Gary W. Adamson, Nov 28 2008
LINKS
N. J. A. Sloane, Transforms
FORMULA
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
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
a(0) = 1; a(n) = Sum_{k=0..floor(n/2)} binomial(n+1,2*k+1) * a(k). - Ilya Gutkovskiy, Apr 07 2022
MAPLE
a:= proc(n) option remember; add(`if`(k<2, 1,
a(iquo(k, 2)))*binomial(n, k), k=0..n)
end:
seq(a(n), n=0..40); # Alois P. Heinz, Jul 08 2015
MATHEMATICA
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 *)
PROG
(PARI) a(n)=1+sum(k=1, n, sum(j=0, n-k, binomial(k, j)*binomial(n-k, j)*a(j)))
(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))
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 22 2004
KEYWORD
easy,nonn,eigen
EXTENSIONS
More terms from Vladeta Jovovic, Jul 26 2002
STATUS
approved