A377857 -id:A377857 - cp's OEIS frontend

A192858 Hosoya indices of the 2n-wheel graphs W_{2n}.

2, 10, 36, 120, 382, 1178, 3550, 10514, 30720, 88788, 254342, 723190, 2043386, 5742490, 16062492, 44744688, 124192270, 343594514, 947857750, 2608015778, 7159034232, 19609583820, 53608363286, 146290947310, 398552156402, 1084153113898, 2944982283540, 7989231439464, 21646950044830, 58585895022218, 158389325993422

Offset: 1

Eric W. Weisstein, Jul 11 2011

Wheel graphs are defined for n >= 4; extended to n=2 using recurrence.

Binomial transform of A120940 multiplied by 2. - R. J. Mathar, Jul 11 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Hosoya Index
Eric Weisstein's World of Mathematics, Wheel Graph
Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1)

I:=[2,10,36,120]; [n le 4 select I[n] else 6*Self(n-1)-11*Self(n-2)+6*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Aug 31 2011

LinearRecurrence[{6, -11, 6, -1}, {10, 36, 120, 382}, {0, 30}]

x='x+O('x^30); Vec(2*x*(1+x)*(x-1)^2/((x^2-3*x+1)^2)) \\ G. C. Greubel, Nov 10 2018

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4).
G.f.: 2*x*(1+x)*(x-1)^2 / ( (x^2 - 3*x + 1)^2 ). - R. J. Mathar, Jul 11 2011
a(n) = ((3+r)^n*((5-r)*n+3*r-5) + (3-r)^n*((5+r)*n-3*r-5))/(5*2^n) with r=sqrt(5). - Bruno Berselli, Aug 31 2011
a(n) = 2*A377857(n+2). - R. J. Mathar, Dec 16 2024

A378383 Number of subwords of the form UDDD in nondecreasing Dyck paths of length 2*n.

Original entry on oeis.org

0, 0, 0, 1, 5, 19, 64, 202, 612, 1803, 5206, 14809, 41650, 116114, 321478, 885169, 2426462, 6627499, 18048088, 49026874, 132901176, 359625015, 971639014, 2621683741, 7065545950, 19022080034, 51163908874, 137499581917, 369235213742, 990822728623, 2657069356996

Offset: 0

Rigoberto Florez, Nov 24 2024

A Dyck path is nondecreasing if the y-coordinates of its valleys form a nondecreasing sequence.

E. Barcucci, A. Del Lungo, S. Fezzi, and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170 (1997), 211-217.
Éva Czabarka, Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerations of peaks and valleys on non-decreasing Dyck paths, Discrete Math., Vol. 341, No. 10 (2018), pp. 2789-2807. See p. 2798.
Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 15, 19.
Rigoberto Flórez, Leandro Junes, and José L. Ramírez, Enumerating several aspects of non-decreasing Dyck paths, Discrete Mathematics, Vol. 342, Issue 11 (2019), 3079-3097. See page 3092.
Index entries for linear recurrences with constant coefficients, signature (10,-39,74,-69,28,-4).

Cf. A000032, A000045, A377679, A377670, A375995, A377857, A377866, A377867.

Table[If[n < 3, 0, (1/5)((n-3)LucasL[2n-5]+LucasL[2n-3]+Fibonacci[2n+2]-5(n+5) 2^(n-4))], {n,0,26}]

a(n) =((n-3)*L(2n-5)+L(2n-3)+F(2n+2) -5*(n+5)*2^(n-4))/5 for n>=3, where F(n) = A000045(n) and L(n) = A000032(n).

G.f.: (-x^5+2 x^4-5 x^3+8 x^2-5 x+1)*x^3/(2 x^3-7 x^2+5 x-1)^2.

cp's OEIS Frontend

A192858 Hosoya indices of the 2n-wheel graphs W_{2n}.

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