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A002728
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Number of n X (n+2) binary matrices.
(Formerly M3593 N1457)
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5
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1, 4, 22, 190, 3250, 136758, 17256831, 7216495370, 10271202313659, 49856692830176512, 826297617412284162618, 46948445432190686211183650, 9200267975562856184153936960940, 6261904454889790650636380541051266410, 14910331834338546882501064075429145637985605
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OFFSET
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0,2
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REFERENCES
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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).
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LINKS
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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]
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FORMULA
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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
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MAPLE
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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+2$2)), s=b(n$2)):
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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