login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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

LINKS

Table of n, a(n) for n=0..53.

Dennis Walsh, Width-restricted finite functions

Donghwi Park, Space-state complexity of Korean chess and Chinese chess, arXiv preprint arXiv:1507.06401, 2015

FORMULA

T(k,n) = n!sum(binomial(k,i)*binomial(i,n-i)*2^(i-n), i=ceiling(n/2)..k). - 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![x^n](1+x+x^2/2)^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(binomial(n,i)*T(j,i)*T(k,n-i),i=0..n). - Dennis P. Walsh, Feb 15 2011

EXAMPLE

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

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 *)

PROG

(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])}

CROSSREFS

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

Sequence in context: A083718 A055948 A071083 * A234574 A010587 A213473

Adjacent sequences:  A141762 A141763 A141764 * A141766 A141767 A141768

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Jul 28 2008

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 3 20:40 EST 2016. Contains 278745 sequences.