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
Andrew Howroyd, Table of n, a(n) for n = 0..50 (terms 0..23 from Alois P. Heinz)
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 *)
PROG
(PARI) a(n) = A(n+1, n) \\ A defined in A028657. - Andrew Howroyd, Mar 01 2023
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Vladeta Jovovic, Feb 04 2000
STATUS
approved