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)

Let R_n(x) be the real and I_n(x) the imaginary part of product(x+I*k,k=0..n). Then, for n=1,2,..., we have R_n(x)=sum((-1)^k*stirling1(n+1,n+1-2*k)*x^(n+1-2*k),k=0..floor((n+1)/2)), I_n(x)=sum((-1)^(k+1)*stirling1(n+1,n-2*k)*x^(n-2*k),k=0..floor(n/2)). - Milan R. Janjic (agnus(AT)blic.net), May 11 2008

Contribution from Kyle Petersen (tkpeters(AT)umich.edu), Oct 15 2008: (Start)

T(n,k) is also the number of permutations of n with "reflection length" k

(i.e., obtained from 12..n by k not necessarily adjacent transpositions).

For example, when n=3, 132, 213, 321 are obtained by one transposition, while

231 and 312 require two transpositions. (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.

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009: (Start)

A000142 equals row sums.

(End)

Sequence in context: A144250 A156367 A008276 this_sequence A143778 A164645 A115755

Adjacent sequences: A094635 A094636 A094637 this_sequence A094639 A094640 A094641

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