%I A094638
%S A094638 1,1,1,1,3,2,1,6,11,6,1,10,35,50,24,1,15,85,225,274,120,1,21,175,735,
%T A094638 1624,1764,720,1,28,322,1960,6769,13132,13068,5040,1,36,546,4536,22449,
%U A094638 67284,118124,109584,40320,1,45,870,9450,63273,269325,723680,1172700
%N A094638 Triangle read by rows: T(n,k) =|s(n,n+1-k)|, where s(n,k) are the signed
Stirling numbers of the first kind (1<=k<=n; in other words, the
unsigned Stirling numbers of the first kind in reverse order).
%C A094638 Triangle of coefficients of the polynomial (x+1)(x+2)...(x+n), expanded
in decreasing powers of x. - T. D. Noe, Feb 22 2008
%C A094638 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
%C A094638 Sum(k*T(n,k), k=1..n) = A121586 - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Aug 14 2006
%C A094638 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
%C A094638 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,...).
%C A094638 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+)! .
%C A094638 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-)! .
%C A094638 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+)!! .
%C A094638 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-)!! .
%C A094638 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+)!!! .
%C A094638 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-)!!! .
%C A094638 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) .
%C A094638 The above results hold for t any real or complex number. (End)
%C A094638 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
%C A094638 Contribution from Kyle Petersen (tkpeters(AT)umich.edu), Oct 15 2008:
(Start)
%C A094638 T(n,k) is also the number of permutations of n with "reflection length"
k
%C A094638 (i.e., obtained from 12..n by k not necessarily adjacent transpositions).
%C A094638 For example, when n=3, 132, 213, 321 are obtained by one transposition,
while
%C A094638 231 and 312 require two transpositions. (End)
%D A094638 E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations
and random generation, Theoretical Computer Science, 159, 1996, 29-42.
%H A094638 T. D. Noe, <a href="b094638.txt">Rows n=1..51 of triangle, flattened</
a>
%H A094638 A. F. Labossiere, <a href="http://members.lycos.co.uk/sobalian/index.html">
Sobalian Coefficients</a>.
%H A094638 A. F. Labossiere, <a href="http://members.lycos.co.uk/stereotomography/
index.html">Miscellaneous</a>.
%H A094638 F. Hivert, J.-C. Novelli and J.-Y. Thibon, <a href="http://arXiv.org/
abs/math.CO/0401089">The Algebra of Binary Search Trees</a>, Theoretical
Computer Science, 339 (2005), 129-165.
%F A094638 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
%e A094638 Triangle starts:
%e A094638 1;
%e A094638 1,1;
%e A094638 1,3,2;
%e A094638 1,6,11,6;
%e A094638 1,10,35,50,24;
%p A094638 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
%Y A094638 Cf. A000108, A014137, A001246, A033536, A000984, A094639, A006134, A082894,
A002897, A079727.
%Y A094638 Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08
2009: (Start)
%Y A094638 A000142 equals row sums.
%Y A094638 (End)
%Y A094638 Sequence in context: A144250 A156367 A008276 this_sequence A143778 A164645
A115755
%Y A094638 Adjacent sequences: A094635 A094636 A094637 this_sequence A094639 A094640
A094641
%K A094638 easy,nonn,tabl
%O A094638 1,5
%A A094638 Andre F. Labossiere (boronali(AT)laposte.net), May 17 2004
%E A094638 Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 14 2006
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