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

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

Alois P. Heinz, Table of n, a(n) for n = 0..1000

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

Crossrefs

Cf. A051164, A051165, A051166, A027826, A100936, A100937, A152198.

Sequence in context: A092247 A331233 A108360 * A051450 A038508 A105695

Adjacent sequences: A051160 A051161 A051162 * A051164 A051165 A051166

Keyword

easy,nonn,eigen

Author

Jonas Wallgren

Extensions

More terms from Vladeta Jovovic, Jul 26 2002

Status

approved