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 A357420 a(n) is the hafnian of the 2n X 2n symmetric matrix defined by M[i,j] = abs(i - j) if min(i, j) < max(i, j) <= 2*min(i, j), and otherwise 0. 0
 1, 1, 1, 8, 86, 878, 13730, 348760, 11622396, 509566864, 26894616012, 1701189027944, 125492778658096, 10738546182981256, 1049631636279244832, 117756049412699967072 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Table of n, a(n) for n=0..15. Wikipedia, Hafnian Wikipedia, Symmetric matrix EXAMPLE a(4) = 86: 0, 1, 0, 0, 0, 0, 0, 0; 1, 0, 1, 2, 0, 0, 0, 0; 0, 1, 0, 1, 2, 3, 0, 0; 0, 2, 1, 0, 1, 2, 3, 4; 0, 0, 2, 1, 0, 1, 2, 3; 0, 0, 3, 2, 1, 0, 1, 2; 0, 0, 0, 3, 2, 1, 0, 1; 0, 0, 0, 4, 3, 2, 1, 0. MATHEMATICA M[i_, j_, n_] := If[Min[i, j] < Max[i, j] <= 2 Min[i, j], Abs[j - i], 0]; a[n_] := Sum[Product[M[Part[PermutationList[s, 2 n], 2 i - 1], Part[PermutationList[s, 2 n], 2 i], 2 n], {i, n}], {s, SymmetricGroup[2 n] // GroupElements}]/(n!*2^n); Array[a, 6, 0] CROSSREFS Cf. A000982 (number of zero matrix elements of M(n)), A003983, A007590 (number of positive matrix elements of M(n)), A049581, A051125, A352967, A353452 (determinant of M(n)), A353453 (permanent of M(n)). Cf. A202038, A336114, A336286, A336400, A338456. Cf. A356481, A356482, A356483, A356484, A357279. Sequence in context: A371897 A180582 A230621 * A371407 A369505 A268052 Adjacent sequences: A357417 A357418 A357419 * A357421 A357422 A357423 KEYWORD nonn,hard,more AUTHOR Stefano Spezia, Sep 27 2022 EXTENSIONS a(6)-a(15) from Pontus von Brömssen, Oct 16 2023 STATUS approved

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Last modified September 7 19:53 EDT 2024. Contains 375749 sequences. (Running on oeis4.)