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A002725 Number of incidence matrices: n X (n+1) binary matrices under row and column permutations.
(Formerly M2957 N1195)
5
1, 3, 13, 87, 1053, 28576, 2141733, 508147108, 402135275365, 1073376057490373, 9700385489355970183, 298434346895322960005291, 31479360095907908092817694945, 11474377948948020660089085281068730, 14568098446466140788730090352230460100956 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

a(0) = 1 by convention.

REFERENCES

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..23

A. Kerber, Experimentelle Mathematik, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]

B. Misek, On the number of classes of strongly equivalent incidence matrices, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.

FORMULA

a(n) = sum_{1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n+1} (fix A[s_1, s_2, ...; t_1, t_2, ...]/(1^s_1*s_1!*2^s_2*s_2!*...*1^t_1*t_1!*2^t_2*t_2!*...)) where fix A[...] = 2^sum_{i, j>=1} (gcd(i, j)*s_i*t_j). - Sean A. Irvine, Jul 31 2014

EXAMPLE

a(1) = 3: [0,0], [0,1], [1,1].

a(2) = 13:

000 000 000 000 001 001 001 001 001 011 011 011 111

000 001 011 111 001 010 011 110 111 011 101 111 111

MAPLE

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},

      {seq(map(p-> p+j*x^i, b(n-i*j, i-1))[], j=0..n/i)}))

    end:

a:= n-> add(add(2^add(add(igcd(i, j)* coeff(s, x, i)*

      coeff(t, x, j), j=1..degree(t)), i=1..degree(s))/

      mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))/

      mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t)),

      t=b(n+1$2)), s=b(n$2)):

seq(a(n), n=0..12);  # Alois P. Heinz, Aug 01 2014

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten @ Table[ Map[ Function[ {p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := Sum[ Sum[ 2^Sum[ Sum [ GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}] / Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}] / Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1,  Exponent[t, x]}], {t, b[n+1, n+1]}], {s,  b[n, n]}]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A002623, A002727, A006148, A002728, A002724.

A diagonal of the array A(m,n) described in A028657. - N. J. A. Sloane, Sep 01 2013

Sequence in context: A001831 A196561 A244755 * A097711 A114477 A116434

Adjacent sequences:  A002722 A002723 A002724 * A002726 A002727 A002728

KEYWORD

nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Feb 04 2000

STATUS

approved

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Last modified January 21 19:42 EST 2019. Contains 319350 sequences. (Running on oeis4.)