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

0,2

Comments

Equals the partial sums of column 3 of triangle A092582.

Links

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

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

Example

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 + ...

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 *)

Prog

(PARI) {a(n)=1+sum(k=4, n+3, k!)*3/4!}
for(n=0, 25, print1(a(n), ", "))
(Magma) [(&+[Factorial(j): j in [2..n+3]])/8: n in [0..30]]; // G. C. Greubel, Sep 05 2022
(SageMath) [sum(factorial(j) for j in (2..n+3))/8 for n in (0..30)] # G. C. Greubel, Sep 05 2022

Crossrefs

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

Cf. A003422, A007489, A092582.

Sequence in context: A306183 A241840 A199318 * A004212 A243241 A060905

Adjacent sequences: A117394 A117395 A117396 * A117398 A117399 A117400

Keyword

nonn

Author

Paul D. Hanna, Mar 11 2006

Status

approved