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a(n) = binomial(2*n+6,n+7)*(n^2+7*n+1)/(n+8) = f(n,n+6) where f is given in A034261.
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%I #15 Nov 10 2017 19:16:11

%S 0,1,19,186,1365,8540,48348,255816,1289739,6273135,29683225,137447310,

%T 625482585,2806282440,12443418600,54633668400,237871030860,

%U 1028260382994,4417404967206,18874729444340,80265980340370,339907420551336,1434074601137640

%N a(n) = binomial(2*n+6,n+7)*(n^2+7*n+1)/(n+8) = f(n,n+6) where f is given in A034261.

%H Michael De Vlieger, <a href="/A034273/b034273.txt">Table of n, a(n) for n = 0..1655</a>

%t f[a_, b_] := Binomial[a + b, b + 1] (a b + a + 1)/(b + 2); Array[f[#, # + 6] &, 22] (* _Michael De Vlieger_, Nov 08 2017 *)

%o (PARI) A034273(n)=binomial(2*n+6, n+7)*(n^2+7*n+1)/(n+8) \\ _M. F. Hasler_, Nov 08 2017

%Y Cf. A034261 (main entry), A034263 - A034275 (other columns and diagonals n -> f(n,n+k)).

%K nonn

%O 0,3

%A _Clark Kimberling_

%E Corrected and extended by _N. J. A. Sloane_, Apr 21 2000

%E Edited by _M. F. Hasler_, Nov 08 2017