A188286 (Signless) coefficient of x^k in the admittance polynomial of the connected antiregular graph A_n.

1, 0, 1, 0, 2, 1, 0, 3, 4, 1, 0, 12, 19, 8, 1, 0, 40, 78, 49, 12, 1, 0, 240, 508, 372, 121, 18, 1, 0, 1260, 2952, 2545, 1056, 226, 24, 1, 0, 10080, 24876, 23312, 10993, 2864, 418, 32, 1, 0, 72576, 190800, 196380, 105460, 32773, 6100, 670, 40, 1, 0, 725760, 1980576, 2154600, 1250980, 433190, 93773, 12800, 1070, 50, 1, 0, 6652800, 18981840, 21989356, 13878120, 5352935, 1331100, 217743, 23280, 1565, 60, 1

Offset

0,5

Links

Table of n, a(n) for n=0..77.

E. Munarini, Characteristic, admittance and matching polynomials of an antiregular graph, Appl. Anal. Discrete Math. 3 (2009), 157-176.

Formula

T(n,k) = sum(|s(n+1,j+1)|*(-1)^(k-j-1)/(ceiling(n/2)^(k-j)),j=0..k-1),

where s(n,k) are the Stirling numbers of the first kind.

Recurrence: ceiling(n+1/2)*T(n+1,k+2) = ceiling(n/2)*(n+1)*T(n,k+2) + (n+1+ceiling(n/2))*T(n,k+1) + T(n,k) - T(n+1,k+1)

Example

Triangle begins:
1
0, 1
0, 2, 1
0, 3, 4, 1
0, 12, 19, 8, 1
0, 40, 78, 49, 12, 1
0, 240, 508, 372, 121, 18, 1
0, 1260, 2952, 2545, 1056, 226, 24, 1
0, 10080, 24876, 23312, 10993, 2864, 418, 32, 1

Mathematica

Flatten[Table[If[n==0&&k==0, 1, If[n==0&&k>0, 0, Sum[Abs[StirlingS1[n+1, j+1]](-1)^(k+j-1)/Ceiling[n/2]^(k-j), {j, 0, k-1}]]], {n, 0, 8}, {k, 0, n}], 1]

Crossrefs

Cf. A132393, A048994.

Sequence in context: A256130 A257566 A345117 * A363154 A101603 A228161

Adjacent sequences: A188283 A188284 A188285 * A188287 A188288 A188289

Keyword

nonn,easy

Author

Emanuele Munarini, Mar 26 2011

Status

approved