Search: id:A039598
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%I A039598
%S A039598 1,2,1,5,4,1,14,14,6,1,42,48,27,8,1,132,165,110,44,10,1,429,572,429,
%T A039598 208,65,12,1,1430,2002,1638,910,350,90,14,1,4862,7072,6188,3808,
%U A039598 1700,544,119,16,1,16796,25194,23256,15504,7752,2907,798,152,18,1
%N A039598 Triangle formed from odd-numbered columns of triangle of expansions of
powers of x in terms of Chebyshev polynomials U_n (x).
%C A039598 T(n,k)=number of leaves at level k+1 in all ordered trees with n+1 edges.
- Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 15 2005
%C A039598 Riordan array ((1-2x-sqrt(1-4x))/(2x^2),(1-2x-sqrt(1-4x))/(2x)). Inverse
array is A053122. - Paul Barry (pbarry(AT)wit.ie), Mar 17 2005
%C A039598 T(n,k)=number of walks of n steps, each in direction N, S,W,or E, starting
at the origin, remaining in the upper half-plane and ending at height
k (see the R. K. Guy reference, p. 5). Example: T(3,2)=6 because
we have ENN, WNN, NEN, NWN, NNE and NNW. - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Apr 15 2005
%C A039598 Triangle T(n,k), 0<=k<=n, read by rows given by : T(0,0)=1, T(n,k)=0
if k<0 or if k>n, T(n,0)=2*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+2*T(n-1,
k)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Mar 30 2007
%C A039598 Number of 2n+1 step walks from (0,0) to (2n+1,2k+1) and consisting of
step u=(1,1) and d=(1,-1) and the path stays in the nonnegative quadrant
. Example : T(2,0)=5 because we have uuudd, uudud, uuddu, uduud,
ududu ; T(2,1)=4 because we have uuuud, uuudu, uuduu, uduuu ; T(2,
2)=1 because we have uuuuu . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Apr 16 2007, Apr 18 2007
%C A039598 Triangle read by rows:T(n,k)=number of lattice paths from (0,0) to (n,
k)that do not go below the line y=0 and consist of steps U=(1,1),
D=(1,-1)and two types of steps H=(1,0); example: T(3,1)=14 because
we have UDU, UUD, 4 HHU paths, 4 HUH paths and 4 UHH paths . - Philippe
DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2007
%C A039598 This triangle belongs to the family of triangles defined by: T(0,0)=1,
T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,
k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing
different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,
2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189;
(1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075;
(2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575;
(3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965;
(3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331;
(5,5) -> A125906 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep
25 2007
%C A039598 With offset [1,1] this is the (ordinary) convolution triangle a(n,m)
with o.g.f. of column nr. m given by (c(x)-1)^m, where c(x) is the
o.g.f. for Catalan numbers A000108. See the Riordan comment by P.
Barry.
%C A039598 T(n, k) is also the number of order-preserving full transformations (of
an n-chain) with exactly k fixed points. [From A. Umar (aumarh(AT)squ.edu.om),
Oct 02 2008]
%C A039598 T(n,k)/2^(2n+1)=coefficients of the maximally flat lowpass digital differentiator
of the order N=2n+3. [From Pavel Holoborodko (pavel(AT)holoborodko.com),
Dec 19 2008]
%D A039598 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions,
National Bureau of Standards Applied Math. Series 55, 1964 (and various
reprintings), p. 796.
%D A039598 B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian),
FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published
by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993;
see p. 29.
%D A039598 Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in
the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article
05.3.7.
%D A039598 W.-J. Woan, L. Shapiro and D. G. Rogers, The Catalan numbers, the Lebesgue
integral and 4^{n-2}, Amer. Math. Monthly, 104 (1997), 926-931.
%D A039598 M. Aigner, Enumeration via ballot numbers, Discrete Math., 308 (2008),
2544-2563.
%D A039598 Higgins, Peter M. Combinatorial results for semigroups of order-preserving
mappings. Math. Proc. Camb. Phil. Soc. 113 (1993), 281-296. [From
A. Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%D A039598 Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving
full transformations. Semigroup Forum 72 (2006), 51-62. [From A.
Umar (aumarh(AT)squ.edu.om), Oct 02 2008]
%H A039598 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National
Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972
[alternative scanned copy].
%H A039598 R. K. Guy, Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs., Vol. 3 (2000), #00.1.6
%F A039598 Row n: C(2n, n-k)-C(2n, n-k-2).
%F A039598 a(n, k) = C(2n+1, n-k)*2*(k+1)/(n+k+2) = A050166(n, n-k) = a(n-1, k-1)+2*a(n-1,
k)+a(n-1, k+1) [with a(0, 0) = 1 and a(n, k) = 0 if n<0 or n0, T(n, k) = Sum_{j=1..n}
T(n-j, k-1)*A000108(j) . G.f. for column k : Sum_{n>=0} T(n, k)*x^n
= x^k*C(x)^(2*k+2) where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f.
for Catalan numbers, A000108 . Sum_{k>=0} T(m, k)*T(n, k) = A000108(m+n+1)
. - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 14 2004
%F A039598 T(n, k) = A009766(n+k+1, n-k) = A033184(n+k+2, 2k+2) . - DELEHAM Philippe
(kolotoko(AT)wanadoo.fr), Feb 14 2004
%F A039598 Sum_{j>=0} T(k, j)*A039599(n-k, j) = A028364(n, k) . - DELEHAM Philippe
(kolotoko(AT)wanadoo.fr), Mar 04 2004
%F A039598 Antidiagonal sum_{k=0..n} T(n-k, k) = A000957(n+3). - Gerald McGarvey
(gerald.mcgarvey(AT)comcast.net), Jun 05 2005
%F A039598 The triangle may also be generated from M^n * [1,0,0,0...], where M =
an infinite tridiagonal matrix with 1's in the super and subdiagonals
and [2,2,2...] in the main diagonal. - Gary W. Adamson (qntmpkt(AT)yahoo.com),
Dec 17 2006
%F A039598 G.f.=G(t,x)=C^2/(1-txC^2), where C=[1-sqrt(1-4x)]/(2x) is the Catalan
function. From here G(-1,x)=C, i.e. the alternating row sums are
the Catalan numbers (A000108). - Emeric Deutsch (deutsch(AT)duke.poly.edu),
Jan 20 2007
%F A039598 Sum_{k, 0<=k<=n}T(n,k)*x^k = A000108(n+1), A001700(n), A049027(n+1),
A076025(n+1), A076026(n+1) for x=0,1,2,3,4 respectively (see square
array in A067345) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Mar 21 2007
%F A039598 Sum_{k, 0<=k<=n}T(n,k)*(k+1)=4^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Mar 30 2007
%F A039598 Sum_{j, j>=0}T(n,j)*binomial(j,k)=A035324(n,k), A035324 with offset 0
(0<=k<=n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 30 2007
%F A039598 T(n,k)=A053121(2*n+1,2*k+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Apr 16 2007, Apr 18 2007
%F A039598 T(n,k) = A039599(n,k)+A039599(n,k+1). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Sep 11 2007
%F A039598 Sum_{k, 0<=k<=n+1}T(n+1,k)*k^2 = A029760(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr),
Dec 16 2007
%F A039598 Sum_{k, 0<=k<=n}T(n,k)*A059841(k)=A000984(n). [From Philippe DELEHAM
(kolotoko(AT)wanadoo.fr), Nov 12 2008]
%F A039598 G.f.: 1/(1-xy-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-....
(continued fraction).
%e A039598 Triangle starts:
%e A039598 1;
%e A039598 2,1;
%e A039598 5,4,1;
%e A039598 14,14,6,1;
%e A039598 42,48,27,8,1;
%Y A039598 Cf. A008313, A039599.
%Y A039598 Row sums : A001700
%Y A039598 Sequence in context: A054456 A096164 A104710 this_sequence A128738 A126181
A154930
%Y A039598 Adjacent sequences: A039595 A039596 A039597 this_sequence A039599 A039600
A039601
%K A039598 nonn,tabl,easy,nice
%O A039598 0,2
%A A039598 N. J. A. Sloane (njas(AT)research.att.com).
%E A039598 More terms from Clark Kimberling (ck6(AT)evansville.edu)
%E A039598 Typo in one entry corrected by Philippe DELEHAM, Dec 16 2007
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