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Riordan array (1-x,x/(1-x)).
3

%I #58 Feb 29 2024 17:21:57

%S 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,

%T 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,

%U 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

%N Riordan array (1-x,x/(1-x)).

%C From _Peter Bala_, Sep 13 2015: (Start)

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

%C 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.

%C 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.

%C (End)

%H Muniru A Asiru, <a href="/A159854/b159854.txt">Table of n, a(n) for n = 0..5151</a>

%H Igor Victorovich Statsenko, <a href="https://aeterna-ufa.ru/sbornik/IN-2024-02-2.pdf#page=15">On the ordinal numbers of triangles of generalized special numbers</a>, Innovation science No 2-2, State Ufa, Aeterna Publishing House, 2024, pp. 15-19. In Russian.

%F 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

%F 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

%F 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

%e Triangle begins:

%e 1

%e -1,1

%e 0,0,1

%e 0,0,1,1

%e 0,0,1,2,1

%e 0,0,1,3,3,1

%e ...

%p seq(seq( binomial(n-2,k-2), k = 0..n), n = 0..12); # _Peter Bala_, Mar 20 2018

%t Table[Binomial[n-2, k-2], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jul 11 2019 *)

%o (Sage) # uses[riordan_array from A256893]

%o riordan_array(1-x, x/(1-x), 8) # _Peter Luschny_, Mar 21 2018

%o (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

%o (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

%Y Cf. A007318, A122433, A159855.

%Y Cf. A144225. - _R. J. Mathar_, Oct 24 2009

%K easy,sign,tabl

%O 0,14

%A _Philippe Deléham_, Apr 24 2009