A080232 Triangle T(n,k) of differences of pairs of consecutive terms of triangle A071919.

1, 1, -1, 1, 0, -1, 1, 1, -1, -1, 1, 2, 0, -2, -1, 1, 3, 2, -2, -3, -1, 1, 4, 5, 0, -5, -4, -1, 1, 5, 9, 5, -5, -9, -5, -1, 1, 6, 14, 14, 0, -14, -14, -6, -1, 1, 7, 20, 28, 14, -14, -28, -20, -7, -1, 1, 8, 27, 48, 42, 0, -42, -48, -27, -8, -1

Offset

0,12

Comments

Row sums are 1,0,0,0,0,0, ... with g.f. 1 = (1-x)^0(1-2x)^0

(1,-1)-Pascal triangle; mirror image of triangle A112467. - Philippe Deléham, Nov 07 2006

Triangle T(n,k), read by rows, given by (1,0,0,0,0,0,0,0,0,...) DELTA (-1,2,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 01 2011

Links

Table of n, a(n) for n=0..65.

T. M. Brown, On the unimodality of convolutions of sequences of binomial coefficients, arXiv:1810.08235 [math.CO] (2018). See p. 8.

Pedro J. Miana, Hideyuki Ohtsuka, Natalia Romero, Sums of powers of Catalan triangle numbers, arXiv:1602.04347 [math.NT], 2016.

Formula

T(n, k) = binomial(n, k) + 2*Sum{j=1...k} (-1)^j binomial(n, k-j).

Sum_{k=0..n} T(n, k)*x^k = (1-x)*(1+x)^(n-1), for n >= 1. - Philippe Deléham, Sep 05 2005

T(n,k) = T(n-1,k-1) + T(n-1,k) with T(n,0)=1, T(n,n)=-1 for n > 0. - Philippe Deléham, Nov 01 2011

T(n,k) =binomial(n-1,k) - binomial(n-1,k-1), for n > 0. T(n,k) = Sum_{i=-k..k} (-1)^i*binomial(n-1,k+i)*binomial(n+1,k-i), for n >= k. T(n,k)=0, for n < k. - Mircea Merca, Apr 28 2012

G.f.: (-1+2*x*y)/(-1+x*y+x). - R. J. Mathar, Aug 11 2015

Example

Rows begin
  1;
  1,  -1;
  1,   0,  -1;
  1,   1,  -1,  -1;
  1,   2,   0,  -2,  -1;
  1,   3,   2,  -2,  -3,  -1;
  1,   4,   5,   0,  -5,  -4,  -1;
  1,   5,   9,   5,  -5,  -9,  -5,  -1;
  1,   6,  14,  14,   0, -14, -14,  -6,  -1;
  1,   7,  20,  28,  14, -14, -28, -20,  -7,  -1;
  1,   8,  27,  48,  42,   0, -42, -48, -27,  -8,  -1;

Maple

T(n, k):=piecewise(n=0, 1, n>0, binomial(n-1, k)-binomial(n-1, k-1)) # Mircea Merca, Apr 28 2012

Crossrefs

Cf. A007318, A071919.

Apart from initial term, same as A037012.

Sequence in context: A175929 A079627 A061398 * A008482 A037012 A112467

Adjacent sequences: A080229 A080230 A080231 * A080233 A080234 A080235

Keyword

easy,sign,tabl

Author

Paul Barry, Feb 09 2003

Status

approved