OFFSET
1,2
LINKS
Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]
Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Bernoulli number.
Wikipedia, Bernoulli number.
Index entries for linear recurrences with constant coefficients, signature (6,-9,4).
FORMULA
a(n) = 2^(2*n - 1) - n - 1.
From Colin Barker, Dec 01 2012: (Start)
a(n) = 6*a(n-1) - 9*a(n-2) + 4*a(n-3).
G.f.: x^2*(2*x - 5)/((x - 1)^2*(4*x - 1)). (End)
EXAMPLE
From Petros Hadjicostas, Jul 25 2020: (Start)
For n = 1, a(1) = 0 because there are no integers i1, i2, i3 >= 1 s.t. i1 + i2 + i3 = 2.
For n = 2, a(2) = 5 because we have 1 + 1 + 2 = 1 + 2 + 1 = 2 + 1 + 1 = 4, multinomial(4, [1, 1, 2]) = multinomial(4, [1, 2, 1]) = multinomial(4, [2, 1, 1]) = 12, B(1) = -1/2, and B(2) = 1/6. Thus, a(2) = (-1/2)*(12*(-1/2) + 12*(-1/2) + 12*(1/6)) = 5. (End)
MATHEMATICA
LinearRecurrence[{6, -9, 4}, {0, 5, 28}, 30] (* Harvey P. Dale, Dec 02 2014 *)
PROG
(PARI) a(n)=2^(2*n-1)-n-1
/* Second PARI program using the definition */
lista(nn) = {my(a=vector(nn)); for(n=1, nn, a[n] = (-1/2)*sum(i1=1, 2*n-2, sum(i2=1, 2*n-i1-1, binomial(2*n, i1)*binomial(2*n-i1, i2)*bernfrac(i1)))); a; } \\ Petros Hadjicostas, Jul 25 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Benoit Cloitre, Nov 30 2006
STATUS
approved