This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A381514 #10 Feb 26 2025 18:12:02
%S A381514 1,2,23,899,85072,15120411,4439935299,1989537541918,1264044973158281,
%T A381514 1090056235155152713,1227540523199054294506
%N A381514 a(n) is the hafnian of a symmetric Toeplitz matrix of order 2*n whose off-diagonal element (i,j) equals the |i-j|-th prime.
%H A381514 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>.
%H A381514 Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>.
%H A381514 Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.
%e A381514 a(2) = 23 because the hafnian of
%e A381514 [d 2 3 5]
%e A381514 [2 d 2 3]
%e A381514 [3 2 d 2]
%e A381514 [5 3 2 d]
%e A381514 equals M_{1,2}*M_{3,4} + M_{1,3}*M_{2,4} + M_{1,4}*M_{2,3} = 2*2 + 3*3 + 5*2 = 23. Here d denotes the generic element on the main diagonal of the matrix from which the hafnian does not depend.
%t A381514 M[i_, j_]:=Prime[Abs[i-j]]; a[n_]:=Sum[Product[M[Part[PermutationList[s, 2n], 2i-1], Part[PermutationList[s, 2n], 2i]], {i, n}], {s, SymmetricGroup[2n]//GroupElements}]/(n!*2^n); Array[a, 5, 0]
%Y A381514 Cf. A374067, A374068.
%Y A381514 Cf. A071078, A071079, A085807, A306457, A318173, A356483.
%K A381514 nonn,hard,more
%O A381514 0,2
%A A381514 _Stefano Spezia_, Feb 25 2025
%E A381514 a(5)-a(10) from _Pontus von Brömssen_, Feb 26 2025