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a(0) = 1; for n > 0, a(n) = (n!*(3*n+1))/2.
1

%I #22 May 03 2022 06:46:12

%S 1,2,7,30,156,960,6840,55440,504000,5080320,56246400,678585600,

%T 8861529600,124540416000,1874333260800,30076510464000,512608352256000,

%U 9247873130496000,176065276907520000,3527707911856128000,74203511249387520000

%N a(0) = 1; for n > 0, a(n) = (n!*(3*n+1))/2.

%H Harry J. Smith, <a href="/A066114/b066114.txt">Table of n, a(n) for n = 0..100</a>

%F G.f.: Sum_{n>=0} (1 + n*x)^(n+1) / (2 + n*x)^(n+1). - _Paul D. Hanna_, Oct 26 2014

%F a(n) = n-th derivative of (1 + 2/x + 3/x^2)/2 at -1. - _Luc Rousseau_, May 03 2022

%e G.f.: A(x) = 1 + 2*x + 7*x^2 + 30*x^3 + 156*x^4 + 960*x^5 + 6840*x^6 + ...

%e where

%e A(x) = 1/2 + (1+x)^2/(2+x)^2 + (1+2*x)^3/(2+2*x)^3 + (1+3*x)^4/(2+3*x)^4 + (1+4*x)^5/(2+4*x)^5 + (1+5*x)^6/(2+5*x)^6 + (1+6*x)^7/(2+6*x)^7 + ...

%t Join[{1},Table[(n!(3n+1))/2,{n,20}]] (* _Harvey P. Dale_, Jun 15 2011 *)

%o (PARI) { for (n=0, 100, a=(n!*(3*n + 1))/2; if (n==0, a=1); write("b066114.txt", n, " ", a) ) } \\ _Harry J. Smith_, Feb 01 2010

%o (PARI) \p200 \\ set precision

%o {A=Vec(sum(n=0, 600, (1.+n*x)^(n+1)/(2 + n*x +O(x^31))^(n+1)) )}

%o for(n=1, #A, print1(round(A[n]), ", ")) \\ _Paul D. Hanna_, Oct 26 2014

%K nonn,easy

%O 0,2

%A _George E. Antoniou_, Dec 05 2001