Search: id:A059260
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%I A059260
%S A059260 1,0,1,1,1,1,0,2,2,1,1,2,4,3,1,0,3,6,7,4,1,1,3,9,13,11,5,1,0,4,
%T A059260 12,22,24,16,6,1,1,4,16,34,46,40,22,7,1,0,5,20,50,80,86,62,29,8,
%U A059260 1,1,5,25,70,130,166,148,91,37,9,1,0,6,30,95,200,296,314,239,128
%N A059260 Triangle read by rows giving coefficient T(i,j) of x^i y^j in 1/(1-y-x*y-x^2) 
               = 1/((1+x)(1-x-y)) for (i,j) = (0,0), (1,0), (0,1), (2,0), (1,1), 
               (0,2), ...
%C A059260 Coefficients of the (left, normalized) shifted cyclotomic polynomial. 
               Or, coefficients of the basic n-th q-series for q=-2. Indeed, let 
               Y_n(x) = sum( x^k, k=0..n), having as roots all the n-th root of 
               unity except 0; then coefficients in x of (-1)^n Y_n(-x-1) gives 
               exactly the n-th row of A059260 and a practical way to compute it. 
               - Olivier Gerard (olivier.gerard(AT)gmail.com), Jul 30 2002
%C A059260 The maximum in the 2n-row is T(n,n) which is A026641; also T(n,n)~2/3*binomial(2n,
               n). The maximum in the (2n-1)-row is T(n-1,n) which is A014300 (but 
               T has not the same definition as in A026637); also T(n-1,n)~1/3*binomial(2n,
               n). Here is a generalization of the formula given in A026641: T(i,
               j)=sum(binomial(i+k-x,j-k)*binomial(j-k+x,k),k=0..j) for all x real 
               (the proof is easy by induction on i+j using T(i,j)=T(i-1,j)+T(i,
               j-1)). - Claude MORIN (claude.morin4(AT)libertysurf.fr), May 21 2002
%C A059260 The second greatest term in the 2n-row is T(n-1,n+1) which is A014301; 
               the second greatest term in the (2n+1)-row is T(n+1,n)=2*T(n-1,n+1) 
               which is 2*A014301. - Claude MORIN.
%C A059260 Diagonal sums give A008346. - Paul Barry (pbarry(AT)wit.ie), Sep 23 2004
%C A059260 Riordan array (1/(1-x^2), 1/(1+x)). As a product of Riordan arrays, factors 
               into the product of (1/(1+x),x) and (1/(1-x),1/(1-x)) (binomial matrix). 
               - Paul Barry (pbarry(AT)wit.ie), Oct 25 2004
%F A059260 G.f.: 1/(1-y-x*y-x^2) = 1 + y + x^2 + xy + y^2 + 2x^2y + 2xy^2 + y^3 
               + ...
%F A059260 T(i, 0) = 1 if i is even or 0 if i is odd, T(0, i) = 1 and otherwise 
               T(i, j) = T(i-1, j) + T(i, j-1); also T(i, j) = sum((-1)^(i+j+m)*binomial(m, 
               j), m=j..i+j). - Robert FERREOL (rferreol(AT)noos.fr), May 17 2002
%F A059260 T(i, j) ~ (i+j)/(2*i+j)*binomial(i+j, j); more precisely, abs(T(i, j)/
               binomial(i+j, j) - (i+j)/(2*i+j) )<=1/(4*(i+j)-2); the proof is by 
               induction on i+j using the formula 2*T(i, j)=binomial(i+j, j)+T(i, 
               j-1). - Claude MORIN
%F A059260 T(n, k)=sum{j=0..n, (-1)^(n-j)binomial(j, k)}. - Paul Barry (pbarry(AT)wit.ie), 
               Aug 25 2004
%F A059260 T(n, k)=sum{j=0..n-k, C(n-j, j)*C(j, n-k-j)} - Paul Barry (pbarry(AT)wit.ie), 
               Jul 25 2005
%F A059260 Equals A097807 * A007318. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Feb 
               21 2007
%F A059260 Equals A128173 * A007318 as infinite lower triangular matrices. - Gary 
               W. Adamson (qntmpkt(AT)yahoo.com), Feb 17 2007
%e A059260 Triangle begins 1; 0,1; 1,1,1; 0,2,2,1; 1,2,4,3,1; ...
%p A059260 read transforms; 1/(1-y-x*y-x^2); SERIES2(%,x,y,12); SERIES2TOLIST(%,
               x,y,12);
%Y A059260 Cf. A059259. Row sums give A001045.
%Y A059260 Seen as a square array read by antidiagonals this is the coefficient 
               of x^k in expansion of 1/((1-x^2)*(1-x)^n) with rows A002620, A002623, 
               A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808 
               etc. (allowing for signs). A058393 would then effectively provide 
               the table for nonpositive n. - Henry Bottomley (se16(AT)btinternet.com), 
               Jun 25 2001
%Y A059260 Cf. A026641, A014300.
%Y A059260 Sequence in context: A034851 A122085 A066287 this_sequence A135229 A081372 
               A101489
%Y A059260 Adjacent sequences: A059257 A059258 A059259 this_sequence A059261 A059262 
               A059263
%K A059260 nonn,tabl,nice
%O A059260 0,8
%A A059260 N. J. A. Sloane (njas(AT)research.att.com), Jan 23 2001

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