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A059516
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Number of different relations between n intervals (possibly of zero length) on a line.
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5
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1, 2, 26, 818, 47834, 4488722, 617364026, 117029670578, 29247820616474, 9318336983422802, 3686400233750527226, 1772940711343815069938, 1018732890334074636017114, 689260602916515788253448082, 542377407779814380777057527226, 491141384755777675851883392430898
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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;
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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