login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A183109 Triangle read by rows: T(n,m) = number of n X m binary matrices with no zero rows or columns (n >= 1, 1 <= m <= n). 12
1, 1, 7, 1, 25, 265, 1, 79, 2161, 41503, 1, 241, 16081, 693601, 24997921, 1, 727, 115465, 10924399, 831719761, 57366997447, 1, 2185, 816985, 167578321, 26666530801, 3776451407065, 505874809287625 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

T(n,m) = T(m,n) is also the number of complete alignments between two strings of sizes m and n, respectively; i.e. the number of complete matchings in a bipartite graph

LINKS

Indranil Ghosh, Rows 1..50, flattened

Ch. A. Charalambides, A problem of arrangements on chessboards and generalizations, Discrete Mathematics 27.2 (1979): 179-186. (Generalizations.)

D. E. Knuth, Problem 11243, Am. Math. Montly 113 (8) (2006) page 759.

John Riordan and Paul R. Stein, Arrangements on chessboards, Journal of Combinatorial Theory, Series A 12.1 (1972): 72-80. See Table page 78.

FORMULA

T(n,m) = Sum_{j=0..m}(-1)^j*C(m, j)*(2^(m-j)-1)^n.

Recursion: T(m,n) = Sum_{k=1..m} T(k,n-1)*C(m,k)*2^k - T(m,n-1).

From Robert FERREOL, Mar 14 2017: (Start)

T(n,m) = Sum_{i = 0 .. n,j = 0 ..m}(-1)^(n+m+i+j)*C(n,i)*C(m,j)*2^(i*j).

Inverse formula of: 2^(n*m) = Sum_{i = 0 .. n , j = 0 ..m} C(n,i)*C(m,j)*T(i,j). (End)

EXAMPLE

Triangle begins

1;

1, 7;

1, 25, 265;

1, 79, 2161, 41503;

1, 241, 16081, 693601, 24997921;

1, 727, 115465, 10924399, 831719761, 57366997447;

1, 2185, 816985, 167578321, 26666530801, 3776451407065, 505874809287625;

...

MAPLE

A183109 := proc(n, m)

    add((-1)^j*binomial(m, j)*(2^(m-j)-1)^n, j=0..m) ;

end proc:

seq(seq(A183109(n, m), m=1..n), n=1..10) ; # R. J. Mathar, Dec 03 2015

MATHEMATICA

Flatten[Table[Sum[ (-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}], {n, 1, 7}, {m, 1, n}]] (* Indranil Ghosh, Mar 14 2017 *)

PROG

(PARI) tabl(nn) = {for(n=1, nn, for(m = 1, n, print1(sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n), ", "); ); print(); ); };

tabl(8); \\ Indranil Ghosh, Mar 14 2017

(Python)

import math

f = math.factorial

def C(n, r): return f(n)/f(r)/f(n - r)

def T(n, m): return sum([(-1)**j*C(m, j)*(2**(m - j) - 1)**n for j in range (0, m+1)])

i=1

for n in range(1, 51):

....for m in range(1, n+1):

........print str(i)+" "+str(T(n, m))

........i+=1 # Indranil Ghosh, Mar 14 2017

CROSSREFS

Cf. A058482 (this gives the general formula, but values only for m=3).

Diagonal gives A048291. Column 2 is A058481.

Sequence in context: A064051 A147385 A147347 * A082172 A053288 A282917

Adjacent sequences:  A183106 A183107 A183108 * A183110 A183111 A183112

KEYWORD

nonn,tabl

AUTHOR

Steffen Eger, Feb 01 2011

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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 28 09:15 EST 2020. Contains 338720 sequences. (Running on oeis4.)