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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,...). 8
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 (list; graph; refs; listen; history; text; internal format)
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

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-Fran├žois 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

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Last modified August 6 10:14 EDT 2020. Contains 336245 sequences. (Running on oeis4.)