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Column 3 of triangle A117396.
2

%I #27 Sep 06 2022 10:28:51

%S 1,4,19,109,739,5779,51139,504739,5494339,65369539,843747139,

%T 11741033539,175200329539,2790549065539,47251477577539,

%U 847548190793539,16053185741897539,320165936763977539,6706533708227657539,147206624680428617539,3378708717041050697539

%N Column 3 of triangle A117396.

%C Equals the partial sums of column 3 of triangle A092582.

%H G. C. Greubel, <a href="/A117397/b117397.txt">Table of n, a(n) for n = 0..445</a>

%F G.f. satisfies A(x) = (1-x)/(1 - 5*x + 5*x^2) * (1 + x^2*A'(x)).

%F a(n) = 1 + Sum_{k=4..n+3} k!*3/4! for n > 0, with a(0)=1.

%F G.f.: W(0)/(8*x*(1-x)) -1/(4*x), where W(k) = 1 + 1/( 1 - x*(k+3)/( x*(k+3) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 20 2013

%F G.f.: (Sum_{n>=0} (n+2)!*x^n)/(8*x*(1-x)) - 1/(4*x). - _Sergei N. Gladkovskii_, Aug 20 2013

%F a(n) = (1/8)*(A007489(n+3) - 1) = (1/8)*(A003422(n+4) - 2). - _G. C. Greubel_, Sep 05 2022

%e G.f.: A(x) = 1 + 4*x + 19*x^2 + 109*x^3 + 739*x^4 + 5779*x^5 + 51139*x^6 + 504739*x^7 + 5494339*x^8 + 65369539*x^9 + 843747139*x^10 + ...

%p a:=n->sum(j!/8,j=2..n): seq(a(n), n=3..21); # _Zerinvary Lajos_, Jan 08 2007

%t Table[Sum[i!/8, {i, 2, n}], {n, 3, 21}] (* _Zerinvary Lajos_, Jul 12 2009 *)

%o (PARI) {a(n)=1+sum(k=4,n+3,k!)*3/4!}

%o for(n=0,25,print1(a(n),", "))

%o (Magma) [(&+[Factorial(j): j in [2..n+3]])/8: n in [0..30]]; // _G. C. Greubel_, Sep 05 2022

%o (SageMath) [sum(factorial(j) for j in (2..n+3))/8 for n in (0..30)] # _G. C. Greubel_, Sep 05 2022

%Y Cf. A117396 (triangle), A014288 (column 1), A056199 (column 2), A003422 (row sums).

%Y Cf. A003422, A007489, A092582.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 11 2006