1,5

Triangle of coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in decreasing powers of x. - T. D. Noe, Feb 22 2008

T(n,k) is the number of deco polyominoes of height n and having k columns. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: T(2,1)=1 and T(2,2)=1 because the deco polyominoes of height 2 are the vertical and horizontal dominoes, having, respectively, 1 and 2 columns. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 14 2006

Sum(k*T(n,k), k=1..n) = A121586 - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 14 2006

Let the triangle U(n,k), 0<=k<=n, read by rows, given by [1,0,1,0,1,0,1,0,1,0,1,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,...] where DELTA is the operator defined in A084938 ; then T(n,k)=U(n-1,k-1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 06 2007

Comments from Tom Copeland (tcjpn(AT)msn.com), Dec 15 2007 (Start): Consider c(t) = column vector(1, t, t^2, t^3, t^4, t^5,...).

Starting at 1 and sampling every integer to the right, we obtain (1,2,3,4,5,...). And T * c(1) = (1, 1*2, 1*2*3, 1*2*3*4,...), giving n! for n>0. Call this sequence the right factorial (n+)! .

Starting at 1 and sampling every integer to the left, we obtain (1,0,-1,-2,-3,-4,-5,...). And T * c(-1) = (1, 1*0, 1*0*-1, 1*0*-1*-2,...) = (1, 0, 0, 0, ...) , the left factorial (n-)! .

Sampling every other integer to the right, we obtain (1,3,5,7,9,...). T * c(2) = (1, 1*3, 1*3*5, ...) = (1,3,15,105,945,...) , giving A001147 for n>0, the right double factorial, (n+)!! .

Sampling every other integer to the left, we obtain (1,-1,-3,-5,-7...). T * c(-2) = (1, 1*-1, 1*-1*-3, 1*-1*-3*-5,...) = (1,-1,3,-15,105,-945,...) = signed A001147, the left double factorial, (n-)!! .

Sampling every 3 steps to the right, we obtain (1,4,7,10,...). T * c(3) = (1, 1*4, 1*4*7,...) = (1,4,28,280,...) , giving n>0, the right triple factorial, (n+)!!! .

Sampling every 3 steps to the left, we obtain (1,-2,-5,-8,-11,...), giving T * c(-3) = (1, 1*-2, 1*-2*-5, 1*-2*-5*-8,...) = (1,-2,10,-80,880,...) = signed A008544, the left triple factorial, (n-)!!! .

The list partition transform A133314 of [1,T * c(t)] gives [1,T * c(-t)] with all odd terms negated; e.g. LPT[1,T*c(2)] = (1,-1,-1,-3,-15,-105,-945,...) = (1,-A001147) . And e.g.f. for [1,T * c(t)] = (1-xt)^(-1/t) .

The above results hold for t any real or complex number. (End)

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

T. D. Noe, Rows n=1..51 of triangle, flattened

A. F. Labossiere, Sobalian Coefficients.

A. F. Labossiere, Miscellaneous.

F. Hivert, J.-C. Novelli and J.-Y. Thibon, The Algebra of Binary Search Trees, Theoretical Computer Science, 339 (2005), 129-165.

With P(n,t) = sum(k=0,...,n-1) T(n,k+1) * t^k = 1*(1+t)*(1+2t)...(1+(n-1)*t) and P(0,t)=1 , exp[P(.,t)*x] = (1-tx)^(-1/t) . T(n,k+1) = (1/k!) (D_t)^k (D_x)^n [ (1-tx)^(-1/t) - 1 ] evaluated at t=x=0 . (1-tx)^(-1/t) - 1 is the e.g.f. for a plane m-ary tree when t= (m-1) . See Bergeron et al. in "Varieties of Increasing Trees". - Tom Copeland (tcjpn(AT)msn.com), Dec 09 2007

Triangle starts:

1;

1,1;

1,3,2;

1,6,11,6;

1,10,35,50,24;

with(combinat): T:=(n, k)->abs(stirling1(n, n+1-k)): for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form - Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 14 2006

Cf. A000108, A014137, A001246, A033536, A000984, A094639, A006134, A082894, A002897, A079727.

Adjacent sequences: A094635 A094636 A094637 this_sequence A094639 A094640 A094641

Sequence in context: A111049 A088617 A008276 this_sequence A115755 A016556 A067050

easy,nonn,tabl

Andre F. Labossiere (boronali(AT)laposte.net), May 17 2004

Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 14 2006