cp's OEIS Frontend

A059516 Number of different relations between n intervals (possibly of zero length) on a line.

%I A059516 #20 Oct 27 2020 09:08:53
%S A059516 1,2,26,818,47834,4488722,617364026,117029670578,29247820616474,
%T A059516 9318336983422802,3686400233750527226,1772940711343815069938,
%U A059516 1018732890334074636017114,689260602916515788253448082,542377407779814380777057527226,491141384755777675851883392430898
%N A059516 Number of different relations between n intervals (possibly of zero length) on a line.
%H A059516 Alois P. Heinz, <a href="/A059516/b059516.txt">Table of n, a(n) for n = 0..224</a>
%H A059516 IBM Ponder This, <a href="https://www.research.ibm.com/haifa/ponderthis/challenges/January2001.html">Jan 01 2001</a>
%F A059516 a(n) = 2*A055203(n)-0^n = Sum_k A059515(k, n).
%F A059516 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
%F A059516 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
%e A059516 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.
%p A059516 b:= proc(n) option remember; `if`(n=0, 1,
%p A059516       add(b(n-j)*binomial(n, j), j=1..n))
%p A059516     end:
%p A059516 a:= n-> add(b(n+k)*binomial(n, k), k=0..n)/2^n:
%p A059516 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 10 2018
%t A059516 T[0, 0] = 1; T[n_, k_] := Sum[(-1)^(k-i) Binomial[k, i] (i(i+1)/2)^n, {i, 0, k}];
%t A059516 a[n_] := Sum[T[n, k], {k, 1, 2n}]; a[0] = 1;
%t A059516 a /@ Range[0, 20] (* _Jean-François Alcover_, Oct 27 2020, from A300729 *)
%Y A059516 Row n=2 of A316674. Row sums of A300729.
%K A059516 nonn,easy
%O A059516 0,2
%A A059516 _Henry Bottomley_, Jan 19 2001