Search: id:A098158
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%I A098158
%S A098158 1,0,1,0,1,1,0,0,3,1,0,0,1,6,1,0,0,0,5,10,1,0,0,0,1,15,15,1,0,0,0,0,7,
%T A098158 35,21,1,0,0,0,0,1,28,70,28,1,0,0,0,0,0,9,84,126,36,1,0,0,0,0,0,1,45,
%U A098158 210,210,45,1,0,0,0,0,0,0,11,165,462,330,55,1,0,0,0,0,0,0,1,66,495,924
%N A098158 Triangle T(n,k) with diagonals T(n,n-k)=binomial(n,2k).
%C A098158 Row sums are A011782. Inverse is A065547.
%C A098158 Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, -1, 0, 0, 0, 
               0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the 
               operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Jul 29 2006
%C A098158 Sum of entries in column k is A001519(k+1)(the odd indexed Fibonacci 
               numbers). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 
               2008]
%H A098158 D. Dumont and J. Zeng, <a href="http://igd.univ-lyon1.fr/home/zeng/public_html/
               paper/publication.html">Polynomes d'Euler et les fractions continues 
               de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.
%F A098158 Triangle T(n, k)=binomial(n, 2(n-k))
%F A098158 Column k is generated by the polynomial sum{j=0..floor(k/2), C(k, 2j)x^(k-j)}. 
               - Paul Barry (pbarry(AT)wit.ie), Jan 22 2005
%F A098158 G.f.: (1-x*y)/((1-x*y)^2 - x^2*y). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Feb 25 2005
%F A098158 Sum_{k, 0<=k<=n}x^k*T(n,k)= A009116(n), A000007(n), A011782(n), A006012(n), 
               A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), 
               A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively 
               . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 04 2006, Oct 15 
               2008, Oct 19 2008
%F A098158 T(n,k)=T(n-1,k-1)+Sum_{i, 0<=i<=k-1}T(n-2-i,k-1-i) ; T(0,0)=1 ; T(n,k)=0 
               if n<0, if k<0, if n<k . E.g. T(8,5)=T(7,4)+T(6,4)+T(5,3)+T(4,2)+T(3,
               1)+T(2,0)=7+15+5+1+0+0=28 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 04 2006
%F A098158 Sum_{k, 0<=k<=n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), 
               A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), 
               A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), 
               A133345(n), A120612(n), A133356(n), A125818(n) for x = 0, 1, 2, 3, 
               4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively 
               . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 24 2007
%F A098158 Sum_{k, 0<=k<=n}T(n,k)*(-x)^(n-k)= A000012(n), A146559(n), A087455(n), 
               A138230(n), A006495(n), A138229(n) for x = 0,1,2,3,4,5 respectively. 
               [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 14 2008]
%F A098158 T(n,k)=A085478(k,n-k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 02 2008]
%e A098158 Rows begin {1}, {0,1}, {0,1,1}, {0,0,3,1}, {0,0,1,6,1},...
%o A098158 (PARI) {T(n,k)=polcoeff(polcoeff((1-x*y)/((1-x*y)^2-x^2*y)+x*O(x^n),n,
               x)+y*O(y^k),k,y)} (Hanna)
%Y A098158 Cf. A098157.
%Y A098158 Cf. A119900 [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 02 2008]
%Y A098158 Sequence in context: A062734 A117389 A122083 this_sequence A110319 A036872 
               A036871
%Y A098158 Adjacent sequences: A098155 A098156 A098157 this_sequence A098159 A098160 
               A098161
%K A098158 easy,nonn,tabl
%O A098158 0,9
%A A098158 Paul Barry (pbarry(AT)wit.ie), Aug 29 2004
%E A098158 Corrected first formula. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Oct 18 2008

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