A141765 Triangle T, read by rows, such that row n equals column 0 of matrix power M^n where M is a triangular matrix defined by M(k+m,k) = binomial(k+m,k) for m=0..2 and zeros elsewhere. Width-2-restricted finite functions.

1, 1, 1, 1, 1, 2, 4, 6, 6, 1, 3, 9, 24, 54, 90, 90, 1, 4, 16, 60, 204, 600, 1440, 2520, 2520, 1, 5, 25, 120, 540, 2220, 8100, 25200, 63000, 113400, 113400, 1, 6, 36, 210, 1170, 6120, 29520, 128520, 491400, 1587600, 4082400, 7484400, 7484400, 1, 7, 49, 336, 2226

Offset: 0

Author

Paul D. Hanna, Jul 28 2008

Keywords

• nonn
• tabf

Comments

T(k,n) is the number of distinct ways in which n labeled objects can be distributed in k labeled urns allowing at most 2 objects to fall in each urn. - N-E. Fahssi, Apr 22 2009

T(k,n) is the number of functions f:[n]->[k] such that the preimage set under f of any element of [k] has size 2 or less. - Dennis P. Walsh, Feb 15 2011

Examples

			This triangle T begins:
1;
1, 1, 1;
1, 2, 4, 6, 6;
1, 3, 9, 24, 54, 90, 90;
1, 4, 16, 60, 204, 600, 1440, 2520, 2520;
1, 5, 25, 120, 540, 2220, 8100, 25200, 63000, 113400, 113400;
1, 6, 36, 210, 1170, 6120, 29520, 128520, 491400, 1587600, 4082400, 7484400, 7484400;
1, 7, 49, 336, 2226, 14070, 83790, 463680, 2346120, 10636920, 42071400, 139708800, 366735600, 681080400, 681080400,
1, 8, 64, 504, 3864, 28560, 201600, 1345680, 8401680, 48444480, 254016000, 1187524800, 4819953600, 16345929600, 43589145600, 81729648000, 81729648000,
1, 9, 81, 720, 6264, 52920, 430920, 3356640, 24811920, 172504080, 1116536400, 6646147200, 35835307200, 171632260800, 711047937600, 2451889440000, 6620101488000, 12504636144000, 12504636144000,
...
Rows 6 and 8 appear in Park (2015). - _N. J. A. Sloane_, Jan 31 2016
Let M be the triangular matrix that begins:
  1;
  1,  1;
  1,  2,  1;
  0,  3,  3,  1;
  0,  0,  6,  4,  1;
  0,  0,  0, 10,  5,  1; ...
where M(k+m,k) = C(k+m,k) for m=0,1,2 and zeros elsewhere.
Illustrate that row n of T = column 0 of M^n for n >= 0 as follows.
The matrix square M^2 begins:
   1;
   2,  1;
   4,  4,  1;
   6, 12,  6,  1;
   6, 24, 24,  8,  1;
   0, 30, 60, 40, 10,  1; ...
with column 0 of M^2 forming row 2 of T.
The matrix cube M^3 begins:
   1;
   3,   1;
   9,   6,   1;
  24,  27,   9,   1;
  54,  96,  54,  12,   1;
  90, 270, 240,  90,  15,   1;
  90, 540, 810, 480, 135,  18,   1; ...
with column 0 of M^3 forming row 3 of T.
T(2,3)=6 because there are 6 ways to lodge 3 distinguishable balls, labeled by numbers 1,2 and 3, in 2 distinguishable boxes, each of which can hold at most 2 balls. - _N-E. Fahssi_, Apr 22 2009
T(5,8)=63000 because there are 63000 ways to assign 8 students to a dorm room when there are 5 different two-bed dorm rooms that are available. (See link for details of the count.) - _Dennis P. Walsh_, Feb 15 2011
		

Links

• Dennis Walsh, Width-restricted finite functions
• Donghwi Park, Space-state complexity of Korean chess and Chinese chess, arXiv preprint arXiv:1507.06401, 2015

Crossrefs

Cf. A003011 (row sums), A000680 (right border); diagonals: A012244, A036774, A003692.

Programs

Maple

seq(seq(n!*sum(binomial(k,j)*binomial(j,n-j)*2^(j-n),j=ceil(n/2)..k),n=0..2*k),k=1..10); # Dennis P. Walsh, Feb 15 2011

Mathematica

T[k_, n_] := If[n == 0, 1, n! Coefficient[(1 + x + x^2/2)^k, x^n]]; TableForm[Table[T[k, n], {k, 0, 10}, {n, 0, 2 k}]] (* N-E. Fahssi, Apr 22 2009 *)

PARI

{T(n,k)=local(M=matrix(n+1,n+1,n,k,if(n>=k,if(n-k<=2,binomial(n-1,k-1))))); if(k>2*n,0,(M^n)[k+1,1])}

Formula

T(k,n) = n!*Sum_{i=ceiling(n/2)..k} binomial(k,i)*binomial(i,n-i)*2^(i-n). - Dennis P. Walsh, Feb 15 2011

T(n,2*n) = (2n)!/2^n; thus the rightmost border of T equals A000680.

Main diagonal (central terms) equals A012244.

Other diagonals include A036774 and A003692.

Row sums of triangle T equals A003011, the number of permutations of up to n kinds of objects, where each kind of object can occur at most two times.

T(k,n) = n^k. Double e.g.f.: Sum_{k,n} T(k,n)*(z^k/k!)*(x^n/n!) = exp(z(1+x+x^2/2)). - N-E. Fahssi, Apr 22 2009

T(j+k,n) = Sum_{i=0..n} binomial(n,i)*T(j,i)*T(k,n-i). - Dennis P. Walsh, Feb 15 2011