OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..224
IBM Ponder This, Jan 01 2001
FORMULA
a(n) = Sum_{m>=0} binomial(m+1,2)^n/2^(m+1). a(n) = (1/2^n)*Sum_{k=0..n} binomial(n,k)*A000670(n+k). - Vladeta Jovovic, Aug 17 2006
E.g.f. as a continued fraction: 1/(1 + 2*(1 - exp(t))/(1 + 2*(1 - exp(2*t))/(1 + 2*(1 - exp(3*t))/(1 + ...)))) = 1 + 2*t + 26*t^2/2! + .... See A300729. - Peter Bala, Jun 13 2019
EXAMPLE
a(1)=2 since if a is starting point of interval and A is end point then possibilities are aA (zero length) or a-A (positive length). a(2)=26 since possibilities are: aAbB, aAb-B, b-aAB, abB-A, a-AbB, ab-AB, aA-bB, bB-aA, aA-b-B, b-aA-B, b-B-aA, bB-a-A, a-bB-A, a-A-bB, ab-A-B, ab-B-A, a-b-AB, b-a-AB, a-bA-B, b-a-AB, a-A-b-B, a-b-A-B, a-b-B-A, b-B-a-A, b-a-B-A, b-a-A-B.
MAPLE
b:= proc(n) option remember; `if`(n=0, 1,
add(b(n-j)*binomial(n, j), j=1..n))
end:
a:= n-> add(b(n+k)*binomial(n, k), k=0..n)/2^n:
seq(a(n), n=0..20); # Alois P. Heinz, Jul 10 2018
MATHEMATICA
T[0, 0] = 1; T[n_, k_] := Sum[(-1)^(k-i) Binomial[k, i] (i(i+1)/2)^n, {i, 0, k}];
a[n_] := Sum[T[n, k], {k, 1, 2n}]; a[0] = 1;
a /@ Range[0, 20] (* Jean-François Alcover, Oct 27 2020, from A300729 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Jan 19 2001
STATUS
approved