OFFSET
1,2
COMMENTS
Eigensequence of a triangle with the natural numbers (1, 2, 3, ...) as the right border, the triangular series (1, 3, 6, ...) as the left border; and the rest zeros. - Gary W. Adamson, Aug 01 2016
LINKS
Harvey P. Dale, Table of n, a(n) for n = 1..449
FORMULA
a(n+1) = k*(k+1)/2 - a(n)*(a(n)+1)/2, where k = a(n) + n + 1.
a(n) = Sum_{i=0..n} (n!/(n-i)!) * (n-i)(n-i+1)/2 = Sum_{i=0..n} (n!/(n-i)!) * A000217(n-i). For n > 2, a(n) = (3*n*(n-1)/2)*floor((n-2)!*e) + n, where e=exp(1). - Max Alekseyev, Feb 14 2005
a(n) = n*a(n-1) + n*(n+1)/2. - Emeric Deutsch, Mar 16 2008
a(n) ~ 3*sqrt(Pi/2)*exp(1)*n^n*sqrt(n)/exp(n). - Ilya Gutkovskiy, Aug 02 2016
E.g.f.: x * (1+x/2) * exp(x) / (1-x). - Seiichi Manyama, Dec 31 2023
EXAMPLE
a(2) = 2 + 3 = 5, a(3) = 6 + 7 + 8 = 21, a(4) = 22 + 23 + 24 + 25.
MAPLE
a[1]:=1: for n from 2 to 20 do a[n]:=n*a[n-1]+(1/2)*n*(n+1) end do: seq(a[n], n=1..20); # Emeric Deutsch, Mar 16 2008
MATHEMATICA
RecurrenceTable[{a[1]==1, a[n]==n*a[n-1]+(n(n+1))/2}, a, {n, 20}] (* Harvey P. Dale, Nov 05 2013 *)
PROG
(PARI) { t(n) = n*(n+1)/2 }
{ a(n) = sum(i=0, n, n!/(n-i)!*t(n-i)) } \\ Max Alekseyev, Feb 14 2005
(PARI) { t(n) = n*(n+1)/2 }
{ a(n) = 3*t(n-1)*floor((n-2)!*exp(1))+n } \\ Max Alekseyev, Feb 14 2005
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Amarnath Murthy, Feb 10 2005
EXTENSIONS
More terms from Max Alekseyev, Feb 14 2005
Name clarified by Seiichi Manyama, Dec 31 2023
STATUS
approved