The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A120305 a(n) = Sum_{i=0..n} Sum_{j=0..n} (-1)^(i+j) * (i+j)!/(i!j!). 14
 1, 1, 3, 9, 31, 111, 407, 1513, 5679, 21471, 81643, 311895, 1196131, 4602235, 17757183, 68680169, 266200111, 1033703055, 4020716123, 15662273839, 61092127491, 238582873475, 932758045123, 3650336341239, 14298633670931 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS p divides a((p+1)/2) for prime p = 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, ... (A033200: primes congruent to {1, 3} mod 8; or, odd primes of the form x^2 + 2*y^2). p divides a((p-3)/2) for prime p = 17, 41, 73, 89, 97, 113, 137, ... (A007519: primes of the form 8n+1). Essentially the same as partial sums of A072547. - Seiichi Manyama, Jan 30 2023 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1664 (terms 0..200 from Vincenzo Librandi) FORMULA a(n) = Sum_{j=0..n} Sum_{i=0..n} (-1)^(i+j)*(i+j)!/(i!j!). Recurrence: 2*n*(3*n-5)*a(n) = 3*(9*n^2 - 19*n + 8)*a(n-1) - 3*(n-1)*(3*n-4)*a(n-2) - 2*(2*n-3)*(3*n-2)*a(n-3). - Vaclav Kotesovec, Aug 13 2013 a(n) ~ 4^(n+1)/(9*sqrt(Pi*n)). - Vaclav Kotesovec, Aug 13 2013 G.f.: ( 1/(sqrt(1-4*x) * (1-x)) ) * ( (1 - x *c(x))/(1 + x *c(x)) ), where c(x) is the g.f. of A000108. - Seiichi Manyama, Jan 30 2023 From Seiichi Manyama, Apr 06 2024: (Start) a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(2*n-3*k-1,n-3*k). a(n) = [x^n] 1/((1+x^3) * (1-x)^n). (End) MATHEMATICA Table[Sum[Sum[(-1)^(i+j)*(i+j)!/(i!j!), {i, 0, n}], {j, 0, n}], {n, 0, 50}] PROG (PARI) a(n) = sum(i=0, n, sum(j=0, n, (-1)^(i+j) * (i+j)!/(i!*j!))); \\ Michel Marcus, Apr 02 2019 (PARI) a(n) = sum(i=0, 2*n, (-1)^i*i!*polcoef(sum(j=0, n, x^j/j!)^2, i)); \\ Seiichi Manyama, May 20 2019 (PARI) my(N=30, x='x+O('x^N)); Vec((1+sqrt(1-4*x))/(sqrt(1-4*x)*(1-x)*(3-sqrt(1-4*x)))) \\ Seiichi Manyama, Jan 30 2023 CROSSREFS Cf. A000108, A006134, A033200, A007519, A092785, A307354. Cf. A072547, A371798. Sequence in context: A151459 A151033 A047012 * A049170 A049174 A209335 Adjacent sequences: A120302 A120303 A120304 * A120306 A120307 A120308 KEYWORD nonn,changed AUTHOR Alexander Adamchuk, Jul 14 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified April 17 01:50 EDT 2024. Contains 371756 sequences. (Running on oeis4.)