The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A381514

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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.
(history; published version)

#10 by Alois P. Heinz at Wed Feb 26 18:12:02 EST 2025

STATUS

proposed

approved

#9 by Pontus von Brömssen at Wed Feb 26 16:11:37 EST 2025

STATUS

editing

proposed

#8 by Pontus von Brömssen at Wed Feb 26 16:11:26 EST 2025

DATA

1, 2, 23, 899, 85072, 15120411, 4439935299, 1989537541918, 1264044973158281, 1090056235155152713, 1227540523199054294506

EXTENSIONS

a(5)-a(910) from Pontus von Brömssen, Feb 26 2025

#7 by Pontus von Brömssen at Wed Feb 26 15:04:21 EST 2025

DATA

1, 2, 23, 899, 85072

1, 2, 23, 899, 85072, 15120411, 4439935299, 1989537541918, 1264044973158281, 1090056235155152713

EXTENSIONS

a(5)-a(9) from Pontus von Brömssen, Feb 26 2025

STATUS

approved

editing

#6 by N. J. A. Sloane at Wed Feb 26 08:58:30 EST 2025

STATUS

proposed

approved

#5 by Stefano Spezia at Wed Feb 26 02:02:59 EST 2025

STATUS

editing

proposed

#4 by Stefano Spezia at Wed Feb 26 02:02:50 EST 2025

CROSSREFS

Cf. A071078, A071079, A085807, A306457, A318173, A356483.

STATUS

proposed

editing

#3 by Stefano Spezia at Tue Feb 25 15:34:19 EST 2025

STATUS

editing

proposed

#2 by Stefano Spezia at Tue Feb 25 15:27:07 EST 2025

NAME

allocated for Stefano Spezia

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.

DATA

1, 2, 23, 899, 85072

OFFSET

0,2

LINKS

Wikipedia, <a href="https://en.wikipedia.org/wiki/Hafnian">Hafnian</a>.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_matrix">Symmetric matrix</a>.

Wikipedia, <a href="http://en.wikipedia.org/wiki/Toeplitz_matrix">Toeplitz Matrix</a>.

EXAMPLE

a(2) = 23 because the hafnian of

[d 2 3 5]

[2 d 2 3]

[3 2 d 2]

[5 3 2 d]

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.

MATHEMATICA

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]

CROSSREFS

Cf. A374067, A374068.

Cf. A071078, A071079, A085807, A306457, A318173.

KEYWORD

allocated

nonn,hard,more

AUTHOR

Stefano Spezia, Feb 25 2025

STATUS

approved

editing

#1 by Stefano Spezia at Tue Feb 25 15:27:07 EST 2025

NAME

allocated for Stefano Spezia

KEYWORD

allocated

STATUS

approved