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A019757
Expansion of 1/((1-5*x)(1-6*x)(1-7*x)).
4
1, 18, 217, 2190, 19981, 170898, 1398097, 11075670, 85654261, 650280378, 4865931577, 35994162750, 263799130141, 1918854559458, 13871649322657, 99770067275430, 714554190919621, 5099624595840138, 36287658424563337, 257574399980365710, 1824480913344734701, 12900603911464394418
OFFSET
0,2
COMMENTS
Column sequence for m=1 (without leading zeros) of the Sheffer triangle (exp(5*x), exp(x)-1) of 5-restricted Stirling2 numbers. See A193685. - Wolfdieter Lang, Oct 07 2011
FORMULA
If we define f(m,j,x) = Sum_{k=j..m} binomial(m,k)*Stirling2(k,j)*x^(m-k) then a(n-2) = f(n,2,5), (n >= 2). - Milan Janjic, Apr 26 2009
E.g.f.: (d^2/dx^2)(exp(5*x)*(exp(x)-1)^2/2!). See the Sheffer triangle comment above. - Wolfdieter Lang, Oct 07 2011
From Vincenzo Librandi, Oct 08 2011: (Start)
a(n) = (7^(n+2) + 5^(n+2) - 2*6^(n+2))/2.
a(n) = 18*a(n-1) - 107*a(n-2) + 210*a(n-3), n >= 3.
a(n) = 13*a(n-1) - 42*a(n-2) + 5^n, a(0)=1, a(1)=18. (End)
E.g.f.: (49*exp(7*x) + 25*exp(5*x) - 72*exp(6*x))/2. - G. C. Greubel, Feb 07 2018
MATHEMATICA
Table[(7^(n+2) + 5^(n+2) - 2*6^(n+2))/2, {n, 0, 30}] (* G. C. Greubel, Feb 07 2018 *)
LinearRecurrence[{18, -107, 210}, {1, 18, 217}, 30] (* Harvey P. Dale, Mar 13 2024 *)
PROG
(Magma) [(7^(n+2)+5^(n+2)-2*6^(n+2))/2: n in [0..20]]; // Vincenzo Librandi, Oct 08 2011
(PARI) for(n=0, 30, print1((7^(n+2) + 5^(n+2) - 2*6^(n+2))/2, ", ")) \\ G. C. Greubel, Feb 07 2018
CROSSREFS
Sequence in context: A111991 A081136 A101188 * A021503 A025470 A046915
KEYWORD
nonn,easy
STATUS
approved