login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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 xrange(n + 1)])

for n in xrange(13): print [T(n, k) for k in xrange(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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified February 20 00:28 EST 2019. Contains 320329 sequences. (Running on oeis4.)