A188286 - OEIS

%I #12 Nov 01 2021 04:56:10

%S 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,

%T 1,0,1260,2952,2545,1056,226,24,1,0,10080,24876,23312,10993,2864,418,

%U 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

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

%H E. Munarini, <a href="https://doi.org/10.2298/AADM0901157M">Characteristic, admittance and matching polynomials of an antiregular graph</a>, Appl. Anal. Discrete Math. 3 (2009), 157-176.

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

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

%F 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)

%e Triangle begins:

%e 1

%e 0, 1

%e 0, 2, 1

%e 0, 3, 4, 1

%e 0, 12, 19, 8, 1

%e 0, 40, 78, 49, 12, 1

%e 0, 240, 508, 372, 121, 18, 1

%e 0, 1260, 2952, 2545, 1056, 226, 24, 1

%e 0, 10080, 24876, 23312, 10993, 2864, 418, 32, 1

%t 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]

%Y Cf. A132393, A048994.

%K nonn,easy

%O 0,5

%A _Emanuele Munarini_, Mar 26 2011