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A103631 Triangle read by rows: T(n,k) = abs(qStirling2(n,k,q)) for q = -1, with 0 <= k <= n. 11
1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 2, 1, 0, 1, 1, 4, 3, 3, 1, 0, 1, 1, 5, 4, 6, 3, 1, 0, 1, 1, 6, 5, 10, 6, 4, 1, 0, 1, 1, 7, 6, 15, 10, 10, 4, 1, 0, 1, 1, 8, 7, 21, 15, 20, 10, 5, 1, 0, 1, 1, 9, 8, 28, 21, 35, 20, 15, 5, 1, 0, 1, 1, 10, 9, 36, 28, 56, 35, 35, 15, 6, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,14
COMMENTS
Previous name: An invertible triangle whose row sums are F(n+1).
Triangle inverse has general term (-1)^(n-k)*binomial(floor(n/2),n-k). Diagonal sums are A103632.
Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 08 2005
Row sums are Fibonacci numbers (A000045).
Another version of triangle in A065941. - Philippe Deléham, Jan 01 2009
From Johannes W. Meijer, Aug 11 2011: (Start)
The T(n,k) coefficients appear in appendix 2 of Parks's remarkable article "A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov" if we assume that the b(r) coefficients are all equal to 1; see the second Maple program.
The T(n,k) triangle is related to a linear (n+1)-th order differential equation with coefficients a(n,k), see triangle A194005.
Parks's triangle appears to be an appropriate name for the triangle given above. (End)
LINKS
Henry W. Gould, A Variant of Pascal's Triangle, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 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) pp. 694-702.
FORMULA
T(n,k) = binomial(floor((2*n-k-1)/2), n-k).
A polynomial recursion which produces this triangle: p(x, n) = p(x, n - 1) + x^2*p(x, n - 2). - Roger L. Bagula, Apr 27 2008
Sum_{k=0..n} T(n,k)*x^k = A152163(n), A000007(n), A000045(n+1), A026597(n), A122994(n+1), A158608(n), A122995(n+1), A158797(n), A122996(n+1), A158798(n), A158609(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Jun 12 2009
G.f.: (1+(y-1)*x)/(1-x-y^2*x^2). - Philippe Deléham, Mar 09 2012
T(n,k) = T(n-1,k) + T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 09 2012
EXAMPLE
From Paul Barry, Oct 02 2009: (Start)
Triangle begins:
1,
0, 1,
0, 1, 1,
0, 1, 1, 1,
0, 1, 1, 2, 1,
0, 1, 1, 3, 2, 1,
0, 1, 1, 4, 3, 3, 1,
0, 1, 1, 5, 4, 6, 3, 1,
0, 1, 1, 6, 5, 10, 6, 4, 1,
0, 1, 1, 7, 6, 15, 10, 10, 4, 1
Production matrix is:
0, 1,
0, 1, 1,
0, 0, 0, 1,
0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 1, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 1 (End)
MAPLE
From Johannes W. Meijer, Aug 11 2011: (Start)
A103631 := proc(n, k): binomial(floor((2*n-k-1)/2), n-k) end: seq(seq(A103631(n, k), k=0..n), n=0..12);
nmax:=12: 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 seq(A103631(n, k), k=0..n) od: seq(seq(A103631(n, k), k=0..n), n=0..nmax); # (End)
MATHEMATICA
p[x, -1] = 0; p[x, 0] = 1; p[x, 1] = x; p[x, 2] = x + x^2; p[x_, n_] := p[x, n] = p[x, n - 1] + x^2*p[x, n - 2]; (* with *) Table[ExpandAll[p[x, n]], {n, 0, 10}]; (* or *) a = Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[a] (* Roger L. Bagula, Apr 27 2008 *)
Table[Binomial[Floor[(2*n - k - 1)/2], n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Aug 27 2016 *)
qStirling2[n_, k_, q_] /; 1 <= k <= n := q^(k - 1) qStirling2[n - 1, k - 1, q] + Sum[q^j, {j, 0, k - 1}] qStirling2[n - 1, k, q];
qStirling2[n_, 0, _] := KroneckerDelta[n, 0];
qStirling2[0, k_, _] := KroneckerDelta[0, k];
qStirling2[_, _, _] = 0;
Table[Abs[qStirling2[n, k, -1]], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 10 2020 *)
PROG
(Haskell)
a103631 n k = a103631_tabl !! n !! k
a103631_row n = a103631_tabl !! n
a103631_tabl = [1] : [0, 1] : f [1] [0, 1] where
f xs ys = zs : f ys zs where
zs = zipWith (+) ([0, 0] ++ xs) (ys ++ [0])
-- Reinhard Zumkeller, May 07 2012
(Magma) /* As triangle: */ [[Binomial(Floor((2*n-k-1)/2), n-k): k in [0..n]]: n in [0..15]]; // Vincenzo Librandi, Aug 28 2016
(Sage)
from sage.combinat.q_analogues import q_stirling_number2
for n in (0..9):
print([abs(q_stirling_number2(n, k).substitute(q=-1)) for k in [0..n]])
# Peter Luschny, Mar 09 2020
CROSSREFS
Cf. A103633 (signed version).
Sequence in context: A198295 A221857 A133607 * A263191 A192517 A309896
KEYWORD
nonn,tabl,easy
AUTHOR
Paul Barry, Feb 11 2005
EXTENSIONS
New name from Peter Luschny, Mar 09 2020
STATUS
approved

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Last modified March 29 02:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)