0,3

S. R. Schwer, Dependances temporelles: les mots pour le dire, Journees Intelligence Artificielle, 1998.

S. R. Schwer, Enumerating and generating Allen's algebra, in preparation

T. D. Noe, Table of n, a(n) for n=0..100

IBM Ponder This, Jan. 2001

a(n) = Sum_(i = 2 to 2n) lambda(i, n), with lambda(p, 1) = if p = 2 then 1 else 0; lambda(p, n) = (p*(p-1)/2)*(lambda(p, n-1)+2*lambda(p-1, n-1)+lambda(p-2, n-1)).

lambda(p, n) = sum_k[( - 1)^(p + k) * C(p, k) * ((k - 1)*k/2)^n]. So if T(m, 0), T(m, 1), ..., T(m, m) is any row of A035317 with m >= 2n - 1 then a(n) = sum_j[( - 1)^j * T(m, j) * ((m - j + 1)*(m - j)/2)^n]; e.g. a(2) = 13 = 1*6^2 - 3*3^2 + 4*1^2 - 2*0^2 = 1*10^2 - 4*6^2 + 7*3^2 - 6*1^2 + 3*0^2 = 1*15^2 - 5*10^2 + 11*6^2 - 13*3^2 + 9*1^2 - 3*0^2 etc. while a(3) = 409 = 1*15^3 - 5*10^3 + 11*6^3 - 13*3^3 + 9*1^3 - 3*0^3 etc. Henry Bottomley (se16(AT)btinternet.com), Jan 03 2001

Row sums of A122193. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 24 2006

a(n) = Sum_{k=0..n} k!*Stirling2(n,k)*A121251(k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 25 2006

E.g.f.: Sum_{m>=0} exp(x*binomial(m,2))/2^(m+1). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Sep 24 2006

a(n) = Sum_{m>=0} binomial(m,2)^n/2^(m+1). a(n) = (1/2^n)*Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A000670(n+k). - Vladeta Jovovic (vladeta(AT)Eunet.yu), Aug 17 2006

In case n = 2 this is the Delannoy number a(2) = D(2,2) = 13.

a(2) = 13 because if you have two intervals [a1,a2] and [b1,b2], using a for a1 or a2 and b for b1 or b2, and writing c if an a is at the same place as a b, we get the following possibilities: aabb, acb, abab, cab, abc, baab, abba, cc, bac, cba, baba, bca, bbaa

lambda := proc(p, n) option remember; if n = 1 then if p = 2 then RETURN(1) else RETURN(0) fi; else RETURN((p*(p-1)/2)*(lambda(p, n-1)+2*lambda(p-1, n-1)+lambda(p-2, n-1))) fi; end; A055203 := n->add(lambda(i, n), i=2..2*n);

Cf. A035317, A055809, A055810.

Sequence in context: A075672 A069876 A126086 this_sequence A088919 A142484 A098890

Adjacent sequences: A055200 A055201 A055202 this_sequence A055204 A055205 A055206

nonn,nice,easy

Sylviane R. Schwer (schwer(AT)lipn.univ-paris13.fr), Jun 22 2000

More terms from Larry Reeves (larryr(AT)acm.org), Oct 04 2000; and from njas Jan 03, 2001.