A192856 Number of matchings in the n-sun graph.

1, 3, 8, 27, 100, 393, 1624, 7017, 31558, 147177, 709592, 3527769, 18045428, 94797147, 510594056, 2815698483, 15877236898, 91442860467, 537363872008, 3219075448251, 19641501806932, 121974079707225, 770381455577048, 4945495555291017, 32249369951426822

Offset

0,2

Comments

Extended to a(0)-a(2) using the sum/recurrence. - Eric W. Weisstein, Oct 03 2017

Links

Andrew Howroyd and Vaclav Kotesovec, Table of n, a(n) for n = 0..780 (terms 3..50 from Andrew Howroyd; terms a(0..2) corrected by Georg Fischer, Jan 20 2019)

Eric Weisstein's World of Mathematics, Independent Edge Set

Eric Weisstein's World of Mathematics, Matching

Eric Weisstein's World of Mathematics, Sun Graph

Formula

a(n) = A000085(n) + 2*(Sum_{ k=1..n } n * binomial(2*n-k-1, k-1) * A000085(n-k) / k). - Andrew Howroyd, Feb 28 2016, corrected by Vaclav Kotesovec, Mar 06 2016

Recurrence (for n>=7): (n-3)*a(n) = 3*(n-3)*a(n-1) + (n^2 - 4*n + 5)*a(n-2) - 3*(n-1)*a(n-3) + (n-1)*a(n-4). - Vaclav Kotesovec, Mar 06 2016

a(n) ~ exp(3*sqrt(n) - n/2 - 13/4) * n^(n/2) / sqrt(2) * (1 + 39/(8*sqrt(n))). - Vaclav Kotesovec, Mar 06 2016

Mathematica

Table[Sum[(2 j - 1)!! Binomial[n, 2 j], {j, 0, n/2}] + 2 Sum[n Binomial[2 n - k - 1, k - 1] Sum[(2 j - 1)!! Binomial[n - k, 2 j], {j, 0, (n - k)/2}]/k, {k, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 06 2016 *)
Join[{1}, RecurrenceTable[{(n - 3) a[n] == 3 (n - 3) a[n - 1] + (n^2 - 4 n
+ 5) a[n - 2] - 3 (n - 1) a[n - 3] + (n - 1) a[n - 4], a[1]==3, a[2]==8, a[3]==27, a[4] == 100}, a, {n, 1, 20}]] (* Eric W. Weisstein, Oct 03 2017, amended by Georg Fischer, Dec 05 2019 *)

Crossrefs

Cf. A000085.

Sequence in context: A145760 A102318 A102206 * A110886 A104854 A226061

Adjacent sequences: A192853 A192854 A192855 * A192857 A192858 A192859

Keyword

nonn

Author

Eric W. Weisstein, Jul 11 2011

Extensions

a(7)-a(20) from Andrew Howroyd, Feb 28 2016

a(0)-a(2) from Eric W. Weisstein, Oct 03 2017

Status

approved