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 A194005 Triangle of the coefficients of an (n+1)-th order differential equation associated with A103631. 9
 1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 6, 3, 1, 1, 6, 5, 10, 6, 4, 1, 1, 7, 6, 15, 10, 10, 4, 1, 1, 8, 7, 21, 15, 20, 10, 5, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1, 1, 10, 9, 36, 28, 56, 35, 35, 15, 6, 1, 1, 11, 10, 45, 36, 84, 56, 70, 35, 21, 6, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS This triangle is a companion to Parks' triangle A103631. The coefficients of triangle A103631(n,k) appear in appendix 2 of Park’s remarkable article “A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov” if we assume that the b(n) coefficients are all equal to 1, see the second Maple program. The a(n,k) coefficients of the triangle given above are related to the coefficients of a linear (n+1)-th order differential equation for the case b(n)=1, see the examples. a(n,k) is also the number of symmetric binary strings of odd length n with Hamming weight k>0 and no consecutive 1's. - Christian Barrientos and Sarah Minion, Feb 27 2018 LINKS Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened Henry W. Gould, A Variant of Pascal's Triangle , The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, p. 257-271. P.C. Parks, A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov , Math. Proc. of the Cambridge Philosophical Society, Vol. 58, Issue 04 (1962) p. 694-702. FORMULA a(n,k) = binomial(floor((2*n+1-k)/2), n-k). a(n,k) = sum(A103631(n1,k), n1=k..n), 0<=k<=n and n>=0. a(n,k) = sum(binomial(floor((2*n1-k-1)/2), n1-k), n1=k..n). T(n,0) = T(n,n) = 1, T(n,k) = T(n-2,k-2) + T(n-1,k), 0 < k < n. - Reinhard Zumkeller, Nov 23 2012 EXAMPLE For the 5th-order linear differential equation the coefficients a(k) are: a(0) = 1, a(1) = a(4,0) = 1, a(2) = a(4,1) = 4, a(3) = a(4,2) = 3, a(4) = a(4,3) = 3 and a(5) = a(4,4) = 1. The corresponding Hurwitz matrices A(k) are, see Parks: A(5) = Matrix([[a(1),a(0),0,0,0], [a(3),a(2),a(1),a(0),0], [a(5),a(4),a(3),a(2),a(1)], [0,0,a(5),a(4),a(3)], [0,0,0,0,a(5)]]), A(4) = Matrix([[a(1),a(0),0,0], [a(3),a(2),a(1),a(0)], [a(5),a(4),a(3),a(2)], [0,0,a(5),a(4)]]), A(3) = Matrix([[a(1),a(0),0], [a(3),a(2),a(1)], [a(5),a(4),a(3)]]), A(2) = Matrix([[a(1),a(0)], [a(3),a(2)]]) and A(1) = Matrix([[a(1)]]). The values of b(k) are, see Parks: b(1) = d(1), b(2) = d(2)/d(1), b(3) = d(3)/(d(1)*d(2)), b(4) = d(1)*d(4)/(d(2)*d(3)) and b(5) = d(2)*d(5)/(d(3)*d(4)). These a(k) values lead to d(k) = 1 and subsequently to b(k) = 1 and this confirms our initial assumption, see the comments. MAPLE A194005 := proc(n, k): binomial(floor((2*n+1-k)/2), n-k) end: for n from 0 to 11 do seq(A194005(n, k), k=0..n) od; seq(seq(A194005(n, k), k=0..n), n=0..11); nmax:=11: for n from 0 to nmax+1 do b(n):=1 od: A103631 := proc(n, k) option remember: local j: if k=0 and n=0 then b(1) elif k=0 and n>=1 then 0 elif k=1 then b(n+1) elif k=2 then b(1)*b(n+1) elif k>=3 then expand(b(n+1)*add(procname(j, k-2), j=k-2..n-2)) fi: end: for n from 0 to nmax do for k from 0 to n do A194005(n, k):= add(A103631(n1, k), n1=k..n) od: od: seq(seq(A194005(n, k), k=0..n), n=0..nmax); MATHEMATICA Flatten[Table[Binomial[Floor[(2n+1-k)/2], n-k], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Apr 15 2012 *) PROG (Haskell) a194005 n k = a194005_tabl !! n !! k a194005_row n = a194005_tabl !! n a194005_tabl = [1] : [1, 1] : f [1] [1, 1] where    f row' row = rs : f row rs where      rs = zipWith (+) ([0, 1] ++ row') (row ++ [0]) -- Reinhard Zumkeller, Nov 22 2012 CROSSREFS Cf. A065941 and A103631. Triangle sums (see A180662): A000071 (row sums; alt row sums), A075427 (Kn22), A000079 (Kn3), A109222(n+1)-1 (Kn4), A000045 (Fi1), A034943 (Ca3), A001519 (Gi3), A000930 (Ze3) Interesting diagonals: T(n,n-4) = A189976(n+5) and T(n,n-5) = A189980(n+6) Cf. A052509. Sequence in context: A243714 A278427 A077592 * A055794 A092905 A052509 Adjacent sequences:  A194002 A194003 A194004 * A194006 A194007 A194008 KEYWORD nonn,easy,tabl AUTHOR Johannes W. Meijer & A. Hirschberg (a.hirschberg(AT)tue.nl), Aug 11 2011 STATUS approved

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Last modified August 18 12:35 EDT 2018. Contains 313832 sequences. (Running on oeis4.)