A117397 Column 3 of triangle A117396.

1, 4, 19, 109, 739, 5779, 51139, 504739, 5494339, 65369539, 843747139, 11741033539, 175200329539, 2790549065539, 47251477577539, 847548190793539, 16053185741897539, 320165936763977539, 6706533708227657539, 147206624680428617539, 3378708717041050697539

Offset: 0

Author

Paul D. Hanna, Mar 11 2006

Keywords

• nonn

Comments

Equals the partial sums of column 3 of triangle A092582.

Examples

			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 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445

Crossrefs

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

Cf. A003422, A007489, A092582.

Programs

Magma

[(&+[Factorial(j): j in [2..n+3]])/8: n in [0..30]]; // G. C. Greubel, Sep 05 2022

Maple

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

Mathematica

Table[Sum[i!/8, {i, 2, n}], {n, 3, 21}] (* Zerinvary Lajos, Jul 12 2009 *)

PARI

{a(n)=1+sum(k=4,n+3,k!)*3/4!}
for(n=0,25,print1(a(n),", "))

SageMath

[sum(factorial(j) for j in (2..n+3))/8 for n in (0..30)] # G. C. Greubel, Sep 05 2022

Formula

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

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

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

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

a(n) = (1/8)*(A007489(n+3) - 1) = (1/8)*(A003422(n+4) - 2). - G. C. Greubel, Sep 05 2022