A159854 Riordan array (1-x,x/(1-x)).

1, -1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 3, 3, 1, 0, 0, 1, 4, 6, 4, 1, 0, 0, 1, 5, 10, 10, 5, 1, 0, 0, 1, 6, 15, 20, 15, 6, 1, 0, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 0, 1, 8, 28, 56, 70, 56, 28, 8, 1, 0, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 0, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1

Offset

0,14

Comments

From Peter Bala, Sep 13 2015: (Start)

The m-th power of the array is the Riordan array (1 - m*x, x/(1 - m*x)).

This array, call it M, is a pseudo-involution in the Riordan group, that is, M*D has order 2, where D = (1,-z) is the diagonal matrix with alternating 1's and -1's on the main diagonal.

This array belongs to the subgroups G := { (f(x)/(x*f'(x)),f(x)): f(x) = x + c(2)*x^2 + c(3)*x^3 + ..., c(i) integral } and H := { (x/f(x),f(x)): f(x) = x + c(2)*x^2 + c(3)*x^3 + ..., c(i) integral } of the Riordan group. Moreover, this array generates the infinite cyclic group (G intersect H). Compare with Pascal's triangle (A007318) which generates the intersection of the Bell subgroup and the hitting-time subgroup of the Riordan group.

(End)

Links

Muniru A Asiru, Table of n, a(n) for n = 0..5151

Igor Victorovich Statsenko, On the ordinal numbers of triangles of generalized special numbers, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

Formula

Exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(x^2/2! + x^3/3!) = x^2/2! + 4*x^3/3! + 10*x^4/4! + 20*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

T(n,k) = C(n,n-k) - 2*C(n-1,n-k-1) + C(n-2,n-k-2), where C(n,k) = n!/(k!*(n-k)!) for 0 <= k <= n, otherwise 0. Cf. A159855. - Peter Bala, Mar 20 2018

T(n,k) = Sum_{i=0..n-k} binomial(n+1, n-k-i)*Stirling2(i + m + 1, i+1) *(-1)^i, where m = 1 for n >= 0, 0 <= k <= n. See A007318, A370516 for m=0 and m=2. - Igor Victorovich Statsenko, Feb 26 2023

Example

Triangle begins:
   1
  -1,1
   0,0,1
   0,0,1,1
   0,0,1,2,1
   0,0,1,3,3,1
...

Maple

seq(seq( binomial(n-2, k-2), k = 0..n), n = 0..12); # Peter Bala, Mar 20 2018

Mathematica

Table[Binomial[n-2, k-2], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 11 2019 *)

Prog

(Sage) # uses[riordan_array from A256893]
riordan_array(1-x, x/(1-x), 8) # Peter Luschny, Mar 21 2018
(GAP) Flat(List([0..12], n->List([0..n], k->Binomial(n, k)-2*Binomial(n-1, n-k-1)+Binomial(n-2, n-k-2)))); # Muniru A Asiru, Mar 22 2018
(Magma) /* As triangle */ [[Binomial(n, n-k)-2*Binomial(n-1, n-k-1)+ Binomial(n-2, n-k-2): k in [0..n]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 11 2019

Crossrefs

Cf. A007318, A122433, A159855.

Cf. A144225. - R. J. Mathar, Oct 24 2009

Sequence in context: A036867 A036866 A144225 * A127840 A017837 A145153

Adjacent sequences: A159851 A159852 A159853 * A159855 A159856 A159857

Keyword

easy,sign,tabl

Author

Philippe Deléham, Apr 24 2009

Status

approved