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%I #126 Apr 15 2024 13:05:00
%S 1,0,1,1,1,1,0,2,2,1,1,2,4,3,1,0,3,6,7,4,1,1,3,9,13,11,5,1,0,4,12,22,
%T 24,16,6,1,1,4,16,34,46,40,22,7,1,0,5,20,50,80,86,62,29,8,1,1,5,25,70,
%U 130,166,148,91,37,9,1,0,6,30,95,200,296,314,239,128,46,10,1
%N 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 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_{k=0..n} x^k, having as roots all the n-th roots of unity except for 0; then coefficients in x of (-1)^n Y_n(-x-1) give exactly the n-th row of A059260 and a practical way to compute it. - _Olivier Gérard_, Jul 30 2002
%C The maximum in the (2n)-th row is T(n,n), which is A026641; also T(n,n) ~ (2/3)*binomial(2n,n). The maximum in the (2n-1)-th row is T(n-1,n), which is A014300 (but T does not have 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_{k=0..j} binomial(i+k-x,j-k)*binomial(j-k+x,k) 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_, May 21 2002
%C The second greatest term in the (2n)-th row is T(n-1,n+1), which is A014301; the second greatest term in the (2n+1)-th row is T(n+1,n) = 2*T(n-1,n+1), which is 2*A014301. - _Claude Morin_
%C Diagonal sums give A008346. - _Paul Barry_, Sep 23 2004
%C Riordan array (1/(1-x^2), x/(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_, Oct 25 2004
%C Signed version is A239473 with relations to partial sums of sequences. - _Tom Copeland_, Mar 24 2014
%C From _Robert Coquereaux_, Oct 01 2014: (Start)
%C Columns of the triangle (cf. Example below) give alternate partial sums along nw-se diagonals of the Pascal triangle, i.e., sequences A000035, A004526, A002620 (or A087811), A002623 (or A173196), A001752, A001753, A001769, A001779, A001780, A001781, A001786, A001808, etc.
%C The dimension of the space of closed currents (distributional forms) of degree p on Gr(n), the Grassmann algebra with n generators, equivalently, the dimension of the space of Gr(n)-valued symmetric multilinear forms with vanishing graded divergence, is V(n,p) = 2^n T(p,n-1) - (-1)^p.
%C If p is odd V(n,p) is also the dimension of the cyclic cohomology group of order p of the Z2 graded algebra Gr(n).
%C If p is even the dimension of this cohomology group is V(n,p)+1.
%C Cf. A193844. (End)
%C From _Peter Bala_, Feb 07 2024: (Start)
%C The following remarks assume the row indexing starts at n = 1.
%C The sequence of row polynomials R(n,x), beginning R(1,x) = 1, R(2,x) = x, R(3,x) = 1 + x + x^2 , ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd( R(n,x), R(m,x)) = R(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the polynomial sequence {R(n,x): n >= 1} is a divisibility sequence; that is, if n divides m then R(n,x) divides R(m,x) in Z[x]. (End)
%H G. C. Greubel, <a href="/A059260/b059260.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H Roland Bacher, <a href="http://arxiv.org/abs/1509.09054">Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle</a>, arXiv:1509.09054 [math.CO], 2015.
%H Joseph Briggs, Alex Parker, Coy Schwieder, and Chris Wells, <a href="https://arxiv.org/abs/2404.07285">Frogs, hats and common subsequences</a>, arXiv:2404.07285 [math.CO], 2024. See p. 28.
%H Robert Coquereaux and Éric Ragoucy, <a href="http://dx.doi.org/10.1016/0393-0440(94)00014-U">Currents on Grassmann algebras</a>, J. of Geometry and Physics, 1995, Vol 15, pp 333-352.
%H Robert Coquereaux and Éric Ragoucy, <a href="http://arxiv.org/abs/hep-th/9310147">Currents on Grassmann algebras</a>, arXiv:hep-th/9310147, 1993.
%H Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 8.
%H Christian Kassel, <a href="http://dx.doi.org/10.1007/BF01459145">A Künneth formula for the cyclic cohomology of Z2-graded algebras</a>, Math. Ann. 275 (1986) 683.
%H Ana Filipa Loureiro and Pascal Maroni, <a href="http://dx.doi.org/10.1007/s11075-012-9573-y">Polynomial sequences associated with the classical linear functionals</a>, Numerical Algorithms, June 2012, Volume 60, Issue 2, pp 297-314. - From _N. J. A. Sloane_, Oct 12 2012
%H Ana Filipa Loureiro and Pascal Maroni, <a href="https://web.archive.org/web/20150913024608/http://cmup.fc.up.pt/cmup/v2/include/filedb.php?id=377&table=publicacoes&field=file">Polynomial sequences associated with the classical linear functionals</a>, preprint, Centro de Matemática da Universidade do Porto.
%H MathOverflow, <a href="http://mathoverflow.net/questions/82560/cyclotomic-polynomials-in-combinatorics">Cyclotomic Polynomials in Combinatorics</a>
%H Mark Norfleet, <a href="https://www.fq.math.ca/Papers1/43-2/paper43-2-12.pdf">Characterization of second-order strong divisibility sequences of polynomials</a>, The Fibonacci Quarterly, 43(2) (2005), 166-169.
%F G.f.: 1/(1-y-x*y-x^2) = 1 + y + x^2 + xy + y^2 + 2x^2y + 2xy^2 + y^3 + ...
%F E.g.f: (exp(-t)+(x+1)*exp((x+1)*t))/(x+2). - _Tom Copeland_, Mar 19 2014
%F O.g.f. (n-th row): ((-1)^n+(x+1)^(n+1))/(x+2). - _Tom Copeland_, Mar 19 2014
%F 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_{m=j..i+j} (-1)^(i+j+m)*binomial(m, j). - _Robert FERREOL_, May 17 2002
%F 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_, May 21 2002
%F T(n, k) = Sum_{j=0..n} (-1)^(n-j)binomial(j, k). - _Paul Barry_, Aug 25 2004
%F T(n, k) = Sum_{j=0..n-k} binomial(n-j, j)*binomial(j, n-k-j). - _Paul Barry_, Jul 25 2005
%F Equals A097807 * A007318. - _Gary W. Adamson_, Feb 21 2007
%F Equals A128173 * A007318 as infinite lower triangular matrices. - _Gary W. Adamson_, Feb 17 2007
%F Equals A130595*A097805*A007318 = (inverse Pascal matrix)*(padded Pascal matrix)*(Pascal matrix) = A130595*A200139. Inverse is A097808 = A130595*(padded A130595)*A007318. - _Tom Copeland_, Nov 14 2016
%F T(i, j) = binomial(i+j, j)-T(i-1, j). - _Laszlo Major_, Apr 11 2017
%F Recurrence for row polynomials (with row indexing starting at n = 1): R(n,x) = x*R(n-1,x) + (x + 1)*R(n-2,x) with R(1,x) = 1 and R(2,x) = x. - _Peter Bala_, Feb 07 2024
%e Triangle begins
%e 1;
%e 0, 1;
%e 1, 1, 1;
%e 0, 2, 2, 1;
%e 1, 2, 4, 3, 1;
%e 0, 3, 6, 7, 4, 1;
%e 1, 3, 9, 13, 11, 5, 1;
%e 0, 4, 12, 22, 24, 16, 6, 1;
%e 1, 4, 16, 34, 46, 40, 22, 7, 1;
%e 0, 5, 20, 50, 80, 86, 62, 29, 8, 1;
%p read transforms; 1/(1-y-x*y-x^2); SERIES2(%,x,y,12); SERIES2TOLIST(%,x,y,12);
%t t[n_, k_] := Sum[ (-1)^(n-j)*Binomial[j, k], {j, 0, n}]; Flatten[ Table[t[n, k], {n, 0, 12}, {k, 0, n}]] (* _Jean-François Alcover_, Oct 20 2011, after _Paul Barry_ *)
%o (Sage)
%o def A059260_row(n):
%o @cached_function
%o def prec(n, k):
%o if k==n: return 1
%o if k==0: return 0
%o return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
%o return [(-1)^(n-k+1)*prec(n+1, n-k+1) for k in (1..n)]
%o for n in (1..9): print(A059260_row(n)) # _Peter Luschny_, Mar 16 2016
%o (PARI) T(n, k) = sum(j=0, n, (-1)^(n - j)*binomial(j, k));
%o for(n=0, 12, for(k=0, n, print1(T(n, k),", ");); print();) \\ _Indranil Ghosh_, Apr 11 2017
%o (Python)
%o from sympy import binomial
%o def T(n, k): return sum((-1)**(n - j)*binomial(j, k) for j in range(n + 1))
%o for n in range(13): print([T(n, k) for k in range(n + 1)]) # _Indranil Ghosh_, Apr 11 2017
%Y Cf. A059259. Row sums give A001045.
%Y 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_, Jun 25 2001
%Y Cf. A026641, A014300.
%Y Cf. A007318, A097805, A097808, A130595, A200139.
%K nonn,tabl,nice,changed
%O 0,8
%A _N. J. A. Sloane_, Jan 23 2001
%E Formula corrected by _Philippe Deléham_, Jan 11 2014
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