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A318943
Number of Dyck paths with n nodes and altitude 3.
2
0, 0, 0, 1, 6, 21, 68, 208, 612, 1752, 4916, 13588, 37128, 100548, 270404, 723208, 1925844, 5110644, 13524872, 35713828, 94140900, 247806600, 651572660, 1711695508, 4493475336, 11789439876, 30917835908, 81053196808, 212426303892, 556607396532
OFFSET
0,5
FORMULA
a(n) = 8*A001906(n+1)-20*A001906(n)-2^(n-5)*(16+3*n), n>=4. - R. J. Mathar, Apr 09 2019
a(n) = 7*a(n-1) - 17*a(n-2) + 16*a(n-3) - 4*a(n-4) for n>7. - Colin Barker, Apr 11 2019
MAPLE
(1-x)^2*x^3*(1+x-3*x^2)/(1-2*x)^2/(1-3*x+x^2) ;
taylor(%, x=0, 30) ;
gfun[seriestolist](%) ; # R. J. Mathar, Nov 25 2018
MATHEMATICA
LinearRecurrence[{7, -17, 16, -4}, {0, 0, 0, 1, 6, 21, 68, 208}, 50] (* Paolo Xausa, May 24 2024 *)
PROG
(PARI) concat([0, 0, 0], Vec(x^3*(1 - x)^2*(1 + x - 3*x^2) / ((1 - 2*x)^2*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Apr 11 2019
CROSSREFS
A column of A318942.
Sequence in context: A123653 A375297 A364636 * A200761 A169687 A302448
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Sep 18 2018
STATUS
approved