A193685 -id:A193685 - cp's OEIS frontend

A019757 Expansion of 1/((1-5x)(1-6x)(1-7*x)).

1, 18, 217, 2190, 19981, 170898, 1398097, 11075670, 85654261, 650280378, 4865931577, 35994162750, 263799130141, 1918854559458, 13871649322657, 99770067275430, 714554190919621, 5099624595840138, 36287658424563337, 257574399980365710, 1824480913344734701, 12900603911464394418

Offset: 0

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (18,-107,210).

[(7^(n+2)+5^(n+2)-2*6^(n+2))/2: n in [0..20]]; // Vincenzo Librandi, Oct 08 2011

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 *)

for(n=0,30, print1((7^(n+2) + 5^(n+2) - 2*6^(n+2))/2, ", ")) \\ G. C. Greubel, Feb 07 2018

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

A143399 Expansion of x^k/Product_{t=k..2k} (1-tx) for k=4.

Original entry on oeis.org

0, 0, 0, 0, 1, 30, 545, 7770, 95781, 1071630, 11192665, 111095490, 1060634861, 9822843030, 88799732385, 787259974410, 6869327386741, 59158464019230, 503954741177705, 4254156112792530, 35637875826743421, 296621138907400230, 2455329298857576625

Offset: 0

Alois P. Heinz, Aug 12 2008

nonn

a(n) is also the number of forests of 4 labeled rooted trees of height at most 1 with n labels, where any root may contain >= 1 labels.

This gives also the fifth column of the Sheffer triangle A143496 (4-restricted Stirling2 numbers). See the e.g.f. given below. See also A193685 for Sheffer comments and the hint for the proof in the o.g.f. formula there. - Wolfdieter Lang, Oct 08 2011

Alois P. Heinz, Table of n, a(n) for n = 0..350
Index entries for sequences related to rooted trees
Index entries for linear recurrences with constant coefficients, signature (30, -355, 2070, -5944, 6720).

4th column of A143395.

a:= proc(k::nonnegint) local M; M := Matrix(k+1, (i,j)-> if (i=j-1) then 1 elif j=1 then [seq(-1* coeff(product(1-t*x, t=k..2*k), x, u), u=1..k+1)][i] else 0 fi); p-> (M^p)[1, k+1] end(4): seq(a(n), n=0..30);

LinearRecurrence[{30,-355,2070,-5944,6720},{0,0,0,0,1},30] (* Harvey P. Dale, Mar 12 2013 *)

G.f.: x^4/((1-4x)(1-5x)(1-6x)(1-7x)(1-8x)).
a(n) = 30a(n-1) -355a(n-2) +2070a(n-3) -5944a(n-4) +6720a(n-5).
E.g.f.: exp(4*x)*((exp(x)-1)^4)/4!. - Wolfdieter Lang, Oct 08 2011

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A019757 Expansion of 1/((1-5x)(1-6x)(1-7*x)).

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A143399 Expansion of x^k/Product_{t=k..2k} (1-tx) for k=4.

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cp's OEIS Frontend

A019757 Expansion of 1/((1-5*x)(1-6*x)(1-7*x)).

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A143399 Expansion of x^k/Product_{t=k..2k} (1-tx) for k=4.

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A019757 Expansion of 1/((1-5x)(1-6x)(1-7*x)).