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 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), ... 24
 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, 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, 130, 166, 148, 91, 37, 9, 1, 0, 6, 30, 95, 200, 296, 314, 239, 128, 46, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS 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 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 Gérard, Jul 30 2002 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_{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 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 Diagonal sums give A008346. - Paul Barry, Sep 23 2004 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 Signed version is A239473 with relations to partial sums of sequences. - Tom Copeland, Mar 24 2014 From Robert Coquereaux, Oct 01 2014: (Start) 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. 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. 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). If p is even the dimension of this cohomology group is V(n,p)+1. Cf. A193844. (End) LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015. R. Coquereaux and E. Ragoucy, Currents on Grassmann algebras, J. of Geometry and Physics, 1995, Vol 15, pp 333-352. R. Coquereaux and E. Ragoucy, Currents on Grassmann algebras, arXiv:hep-th/9310147, 1993. C. Kassel, A Künneth formula for the cyclic cohomology of Z2-graded algebras, Math.  Ann. 275 (1986) 683. Ana Filipa Loureiro and Pascal Maroni, Polynomial sequences associated with the classical linear functionals, Numerical Algorithms, June 2012, Volume 60, Issue 2, pp 297-314. - From N. J. A. Sloane, Oct 12 2012 Ana Filipa Loureiro and Pascal Maroni, Polynomial sequences associated with the classical linear functionals, preprint, Centro de Matemática da Universidade do Porto. MathOverflow, Cyclotomic Polynomials in Combinatorics FORMULA G.f.: 1/(1-y-x*y-x^2) = 1 + y + x^2 + xy + y^2 + 2x^2y + 2xy^2 + y^3 + ... E.g.f: (exp(-t)+(x+1)*exp((x+1)*t))/(x+2). - Tom Copeland, Mar 19 2014 O.g.f. (n-th row): ((-1)^n+(x+1)^(n+1))/(x+2). - Tom Copeland, Mar 19 2014 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 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 T(n, k) = Sum_{j=0..n} (-1)^(n-j)binomial(j, k). - Paul Barry, Aug 25 2004 T(n, k) = Sum_{j=0..n-k} binomial(n-j, j)*binomial(j, n-k-j). - Paul Barry, Jul 25 2005 Equals A097807 * A007318. - Gary W. Adamson, Feb 21 2007 Equals A128173 * A007318 as infinite lower triangular matrices. - Gary W. Adamson, Feb 17 2007 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 T(i, j) = binomial(i+j, j)-T(i-1, j). - Laszlo Major, Apr 11 2017 EXAMPLE Triangle begins   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, 24, 16,  6,  1;   1,  4, 16, 34, 46, 40, 22,  7,  1;   0,  5, 20, 50, 80, 86, 62, 29,  8,  1; MAPLE read transforms; 1/(1-y-x*y-x^2); SERIES2(%, x, y, 12); SERIES2TOLIST(%, x, y, 12); MATHEMATICA 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 *) PROG (Sage) def A059260_row(n):     @cached_function     def prec(n, k):         if k==n: return 1         if k==0: return 0         return -prec(n-1, k-1)-sum(prec(n, k+i-1) for i in (2..n-k+1))     return [(-1)^(n-k+1)*prec(n+1, n-k+1) for k in (1..n)] for n in (1..9): print(A059260_row(n)) # Peter Luschny, Mar 16 2016 (PARI) T(n, k) = sum(j=0, n, (-1)^(n - j)*binomial(j, k)); for(n=0, 12, for(k=0, n, print1(T(n, k), ", "); ); print(); ) \\ Indranil Ghosh, Apr 11 2017 (Python) from sympy import binomial def T(n, k): return sum((-1)**(n - j)*binomial(j, k) for j in range(n + 1)) for n in range(13): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 11 2017 CROSSREFS Cf. A059259. Row sums give A001045. 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 Cf. A026641, A014300. Cf. A007318, A097805, A097808, A130595, A200139. Sequence in context: A209805 A238453 A066287 * A239473 A135229 A257543 Adjacent sequences:  A059257 A059258 A059259 * A059261 A059262 A059263 KEYWORD nonn,tabl,nice AUTHOR N. J. A. Sloane, Jan 23 2001 EXTENSIONS Formula corrected by Philippe Deléham, Jan 11 2014 STATUS approved

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Last modified April 17 10:49 EDT 2021. Contains 343064 sequences. (Running on oeis4.)