%I M3593 N1457 #39 Mar 01 2023 11:23:56
%S 1,4,22,190,3250,136758,17256831,7216495370,10271202313659,
%T 49856692830176512,826297617412284162618,46948445432190686211183650,
%U 9200267975562856184153936960940,6261904454889790650636380541051266410,14910331834338546882501064075429145637985605
%N Number of n X (n+2) binary matrices.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Andrew Howroyd, <a href="/A002728/b002728.txt">Table of n, a(n) for n = 0..50</a> (terms 0..23 from Alois P. Heinz)
%H A. Kerber, <a href="/A002727/a002727.pdf">Experimentelle Mathematik</a>, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 19 (1988), 77-83. [Annotated scanned copy]
%H B. Misek, <a href="http://dml.cz/dmlcz/108444">On the number of classes of strongly equivalent incidence matrices</a>, (Czech with English summary) Casopis Pest. Mat. 89 1964 211-218.
%F a(n) = sum {1*s_1+2*s_2+...=n, 1*t_1+2*t_2+...=n+2} (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
%p b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
%p {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
%p end:
%p a:= n-> add(add(2^add(add(igcd(i, j)* coeff(s, x, i)*
%p coeff(t, x, j), j=1..degree(t)), i=1..degree(s))/
%p mul(i^coeff(s, x, i)*coeff(s, x, i)!, i=1..degree(s))/
%p mul(i^coeff(t, x, i)*coeff(t, x, i)!, i=1..degree(t)),
%p t=b(n+2$2)), s=b(n$2)):
%p seq(a(n), n=0..12); # _Alois P. Heinz_, Aug 01 2014
%t b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Table[Function[{p}, p + j*x^i]@ b[n-i*j, i-1] , {j, 0, n/i}]]] // Flatten; 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+2, n+2]}], {s, b[n, n]}]; Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Nov 28 2014, after _Alois P. Heinz_ *)
%o (PARI) a(n) = A(n+2,n) \\ A defined in A028657. - _Andrew Howroyd_, Mar 01 2023
%Y Cf. A002623, A002727, A006148, A002724, A002725.
%Y A diagonal of the array A(m,n) described in A028657. - _N. J. A. Sloane_, Sep 01 2013
%K nonn
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Feb 04 2000