A003470 a(n) = n*a(n-1) - a(n-2) + 1 + (-1)^n. (Formerly M2759)

1, 1, 3, 8, 31, 147, 853, 5824, 45741, 405845, 4012711, 43733976, 520795003, 6726601063, 93651619881, 1398047697152, 22275111534553, 377278848390249, 6768744159489931, 128228860181918440, 2557808459478878871, 53585748788874537851, 1176328664895760953853

Offset

0,3

Comments

Row sums of A086764. - Philippe Deléham, Apr 27 2004

a(n+2m) == a(n) (mod m) for all n and m. - Robert Israel, Dec 06 2016

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..100

John Riordan and N. J. A. Sloane, Correspondence, 1974

Formula

Diagonal sums of reverse of permutation triangle A008279. a(n) = Sum_{k=0..floor(n/2)} (n-k)!/k!. - Paul Barry, May 12 2004

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*(n-2k)!. - Paul Barry Dec 15 2010

G.f.: 1/(1-x^2-x/(1-x/(1-x^2-2x/(1-2x/(1-x^2-3x/(1-3x/(1-x^2-4x/(1-4x/(1-.... (continued fraction);

G.f.: 1/(1-x-x^2-x^2/(1-3x-x^2-4x^2/(1-5x-x^2-9x^2/(1-7x-x^2-16x^2/(1-... (continued fraction). - Paul Barry, Dec 15 2010

G.f.: hypergeom([1,1],[],x/(1-x^2))/(1-x^2). - Mark van Hoeij, Nov 08 2011

G.f.: 1/Q(0), where Q(k)= 1 - x^2 - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 20 2013

From Robert Israel, Dec 06 2016: (Start)

a(2m) = hypergeom([1,-m,m+1],[],-1).

a(2m+1) = hypergeom([1,-m,m+2],[],-1)*(m+1).

a(2m-1) + a(2m+1) = (2m+1) a(2m). (End)

0 = a(n)*(-a(n+2) - a(n+3)) + a(n+1)*(-2 + a(n+1) - 2*a(n+3) + a(n+4)) + a(n+2)*(-2*a(n+3) + a(n+4)) + a(n+3)*(+2 - a(n+3)) if n >= 0. - Michael Somos, Dec 06 2016

0 = a(n)*(-a(n+2) + a(n+4)) + a(n+1)*(+a(n+1) - a(n+2) - a(n+3) + 3*a(n+4) - a(n+5)) + a(n+2)*(-a(n+3) + a(n+4)) + a(n+3)*(-a(n+4) + a(n+5)) + a(n+4)*(-a(n+4)) if n >= 0. - Michael Somos, Dec 06 2016

a(n) = Sum_{k=0..n} (-1)^k*hypergeom([k+1, k-n], [], -1). - Peter Luschny, Oct 05 2017

D-finite with recurrence: a(n) -n*a(n-1) +(n-2)*a(n-3) -a(n-4)=0. - R. J. Mathar, Apr 29 2020

a(n) ~ n! * (1 + 1/n + 1/(2*n^2) + 2/(3*n^3) + 25/(24*n^4) + 77/(40*n^5) + 2971/(720*n^6) + 6287/(630*n^7) + 1074809/(40320*n^8) + 28160749/(362880*n^9) + ...). - Vaclav Kotesovec, Nov 25 2022

Example

G.f. = 1 + x + 3*x^2 + 8*x^3 + 31*x^4 + 147*x^5 + 853*x^6 + 5824*x^7 + ...

Maple

f:= gfun:-rectoproc({a(n) -(n-1)*a(n-1)-(n-2)*a(n-2)+a(n-3)-2=0, a(0)=1, a(1)=1, a(2)=3}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Dec 06 2016

Mathematica

t = {1, 1}; Do[AppendTo[t, n*t[[-1]] - t[[-2]] + 1 + (-1)^n], {n, 2, 20}] (* T. D. Noe, Oct 07 2013 *)
T[n_, k_] := HypergeometricPFQ[{k+1, k-n}, {}, -1];
Table[Sum[(-1)^k T[n, k], {k, 0, n}], {n, 0, 22}] (* Peter Luschny, Oct 05 2017 *)

Crossrefs

Cf. A008279, A072374, A086764, A143409.

Sequence in context: A213092 A352624 A108492 * A176304 A368208 A180385

Adjacent sequences: A003467 A003468 A003469 * A003471 A003472 A003473

Keyword

nonn,easy

Author

N. J. A. Sloane and John Riordan

Extensions

More terms from Gabriel Cunningham (gcasey(AT)mit.edu), Oct 25 2004

Status

approved