This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A059260 #158 Aug 15 2025 22:28:44
%S A059260 1,0,1,1,1,1,0,2,2,1,1,2,4,3,1,0,3,6,7,4,1,1,3,9,13,11,5,1,0,4,12,22,
%T A059260 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 A059260 130,166,148,91,37,9,1,0,6,30,95,200,296,314,239,128,46,10,1
%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_{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 A059260 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 A059260 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 A059260 Diagonal sums give A008346. - _Paul Barry_, Sep 23 2004
%C A059260 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 A059260 Signed version is A239473 with relations to partial sums of sequences. - _Tom Copeland_, Mar 24 2014
%C A059260 From _Robert Coquereaux_, Oct 01 2014: (Start)
%C A059260 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 A059260 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 A059260 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 A059260 If p is even the dimension of this cohomology group is V(n,p)+1.
%C A059260 Cf. A193844. (End)
%C A059260 From _Peter Bala_, Feb 07 2024: (Start)
%C A059260 The following remarks assume the row indexing starts at n = 1.
%C A059260 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)
%C A059260 From _Miquel A. Fiol_, Oct 04 2024: (Start)
%C A059260 For j>=1, T(i,j) is the independence number of the (i-j)-supertoken graph FF_(i-j)(S_j) of the star graph S_j with j points.
%C A059260 (Given a graph G on n vertices and an integer k>=1, the k-supertoken (or reduced k-th power) FF_k(G) of G has vertices representing configurations of k indistinguishable tokens in the (not necessarily different) vertices of G, with two configurations being adjacent if one can be obtained from the other by moving one token along an edge. See an example below.)
%C A059260 Following the suggestion of Peter Munn, the k-supertoken graph FF_k(S_j) can also be defined as follows: Consider the Lattice graph L(k,j), whose vertices are the k^j j-vectors with elements in the set {0,..,k-1}, two being adjacent if they differ in just one coordinate by one unity. Then, FF_k(S_j) is the subgraph of L(k+1,j) induced by the vertices at distance at most k from (0,..,0). (End)
%H A059260 G. C. Greubel, <a href="/A059260/b059260.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A059260 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 A059260 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 A059260 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 A059260 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 A059260 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 A059260 R. H. Hammack and G. D. Smith, <a href="https://doi.org/10.26493/1855-3974.856.4d2"> Cycle bases of reduced powers of graphs</a>, Ars Math. Contemp. 12 (2017) 183-203.
%H A059260 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 A059260 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 A059260 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 A059260 MathOverflow, <a href="http://mathoverflow.net/questions/82560/cyclotomic-polynomials-in-combinatorics">Cyclotomic Polynomials in Combinatorics</a>
%H A059260 Mark Norfleet, <a href="https://web.archive.org/web/2024*/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 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 E.g.f: (exp(-t)+(x+1)*exp((x+1)*t))/(x+2). - _Tom Copeland_, Mar 19 2014
%F A059260 O.g.f. (n-th row): ((-1)^n+(x+1)^(n+1))/(x+2). - _Tom Copeland_, Mar 19 2014
%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_{m=j..i+j} (-1)^(i+j+m)*binomial(m, j). - _Robert FERREOL_, 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_, May 21 2002
%F A059260 T(n, k) = Sum_{j=0..n} (-1)^(n-j)binomial(j, k). - _Paul Barry_, Aug 25 2004
%F A059260 T(n, k) = Sum_{j=0..n-k} binomial(n-j, j)*binomial(j, n-k-j). - _Paul Barry_, Jul 25 2005
%F A059260 Equals A097807 * A007318. - _Gary W. Adamson_, Feb 21 2007
%F A059260 Equals A128173 * A007318 as infinite lower triangular matrices. - _Gary W. Adamson_, Feb 17 2007
%F A059260 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 A059260 T(i, j) = binomial(i+j, j)-T(i-1, j). - _Laszlo Major_, Apr 11 2017
%F A059260 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
%F A059260 From _Miquel A. Fiol_, Sep 30 2024: (Start)
%F A059260 The triangle can be seen as a slice of a 3-dimensional table that links it to well-known sequences as follows.
%F A059260 The j-th column of the triangle, T(i,j) for i >= j, equals A(n,c1,c2) = Sum_{k=0..floor(n/2)} binomial(c1+2*k-1,2*k)*binomial(c2+n-2*k-1,n-2*k) when c1=1, c2=j, and n=i-j.
%F A059260 This gives T(i,j) = Sum_{k=0..floor((i-j)/2)} binomial(i-2*k-1, j-1). For other values of (c1,c2), see the example below. (End)
%e A059260 Triangle begins
%e A059260 1;
%e A059260 0, 1;
%e A059260 1, 1, 1;
%e A059260 0, 2, 2, 1;
%e A059260 1, 2, 4, 3, 1;
%e A059260 0, 3, 6, 7, 4, 1;
%e A059260 1, 3, 9, 13, 11, 5, 1;
%e A059260 0, 4, 12, 22, 24, 16, 6, 1;
%e A059260 1, 4, 16, 34, 46, 40, 22, 7, 1;
%e A059260 0, 5, 20, 50, 80, 86, 62, 29, 8, 1;
%e A059260 Sequences obtained with _Miquel A. Fiol_'s Sep 30 2024 formula of A(n,c1,c2) for other values of (c1,c2). (In the table, rows are indexed by c1=0..6 and columns by c2=0..6):
%e A059260 A000007 A000012 A000027 A025747 A000292* A000332* A000389*
%e A059260 A059841 A008619 A087811* A002623 A001752 A001753 A001769
%e A059260 A193356 A008794* A005993 A005994 ------- ------- -------
%e A059260 ------- ------- ------- A005995 A018210 ------- A052267
%e A059260 ------- ------- ------- ------- A018211 A018212 -------
%e A059260 ------- ------- ------- ------- ------- A018213 A018214
%e A059260 ------- ------- ------- ------- ------- ------- A062136
%e A059260 *requires offset adjustment.
%e A059260 The 2-supertoken FF_2(S_3) of the star graph S_3 with central vertex 1 and peripheral vertices 2,3,4. (The vertex `ij' of FF_2(S_3) represents the configuration of one token in `ì' and the other token in `j'). The T(5,3)=7 independent vertices are 22, 24, 44, 23, 11, 34, and 33.
%e A059260 22--12---24---14---44
%e A059260 | \ / |
%e A059260 23 11 34
%e A059260 \ | /
%e A059260 13
%e A059260 |
%e A059260 33
%p A059260 read transforms; 1/(1-y-x*y-x^2); SERIES2(%,x,y,12); SERIES2TOLIST(%,x,y,12);
%t A059260 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 A059260 (Sage)
%o A059260 def A059260_row(n):
%o A059260 @cached_function
%o A059260 def prec(n, k):
%o A059260 if k==n: return 1
%o A059260 if k==0: return 0
%o A059260 return -prec(n-1,k-1)-sum(prec(n,k+i-1) for i in (2..n-k+1))
%o A059260 return [(-1)^(n-k+1)*prec(n+1, n-k+1) for k in (1..n)]
%o A059260 for n in (1..9): print(A059260_row(n)) # _Peter Luschny_, Mar 16 2016
%o A059260 (PARI) T(n, k) = sum(j=0, n, (-1)^(n - j)*binomial(j, k));
%o A059260 for(n=0, 12, for(k=0, n, print1(T(n, k),", ");); print();) \\ _Indranil Ghosh_, Apr 11 2017
%o A059260 (Python)
%o A059260 from sympy import binomial
%o A059260 def T(n, k): return sum((-1)**(n - j)*binomial(j, k) for j in range(n + 1))
%o A059260 for n in range(13): print([T(n, k) for k in range(n + 1)]) # _Indranil Ghosh_, Apr 11 2017
%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_, Jun 25 2001
%Y A059260 Cf. A026641, A014300.
%Y A059260 Cf. A007318, A097805, A097808, A130595, A200139, A062135.
%K A059260 nonn,tabl,nice
%O A059260 0,8
%A A059260 _N. J. A. Sloane_, Jan 23 2001
%E A059260 Formula corrected by _Philippe Deléham_, Jan 11 2014