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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.

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.

Index entries for triangles and arrays related to Pascal's triangle

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 February 25 00:39 EST 2018. Contains 299630 sequences. (Running on oeis4.)