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Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).
77

%I #55 Nov 23 2023 07:16:32

%S 1,0,1,0,1,1,0,0,3,1,0,0,1,6,1,0,0,0,5,10,1,0,0,0,1,15,15,1,0,0,0,0,7,

%T 35,21,1,0,0,0,0,1,28,70,28,1,0,0,0,0,0,9,84,126,36,1,0,0,0,0,0,1,45,

%U 210,210,45,1,0,0,0,0,0,0,11,165,462,330,55,1,0,0,0,0,0,0,1,66,495,924

%N Triangle T(n,k) with diagonals T(n,n-k) = binomial(n, 2*k).

%C Row sums are A011782. Inverse is A065547.

%C Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Jul 29 2006

%C Sum of entries in column k is A001519(k+1) (the odd-indexed Fibonacci numbers). - _Philippe Deléham_, Dec 02 2008

%C Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with k left-to-right minima. A left-to-right minimum in a permutation a(1)a(2)...a(n) is position i such that a(j) > a(i) for all j < i. - _Tian Han_, Nov 16 2023

%H G. C. Greubel, <a href="/A098158/b098158.txt">Rows n = 0..49 of triangle, flattened</a>

%H D. Dumont and J. Zeng, <a href="http://math.univ-lyon1.fr/homes-www/zeng/public_html/paper/publication.html">Polynomes d'Euler et les fractions continues de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.

%H Tian Han and Sergey Kitaev, <a href="https://arxiv.org/abs/2311.02974">Joint distributions of statistics over permutations avoiding two patterns of length 3</a>, arXiv:2311.02974 [math.CO], 2023.

%F T(n,k) = binomial(n,2*(n-k)).

%F From _Tom Copeland_, Oct 10 2016: (Start)

%F E.g.f.: exp(t*x) * cosh(t*sqrt(x)).

%F O.g.f.: (1/2) * ( 1 / (1 - (1 + sqrt(1/x))*x*t) + 1 / (1 - (1 - sqrt(1/x))*x*t) ).

%F Row polynomial: x^n * ((1 + sqrt(1/x))^n + (1 - sqrt(1/x))^n) / 2. (End)

%F Column k is generated by the polynomial Sum_{j=0..floor(k/2)} C(k, 2j) * x^(k-j). - _Paul Barry_, Jan 22 2005

%F G.f.: (1-x*y)/((1-x*y)^2 - x^2*y). - _Paul D. Hanna_, Feb 25 2005

%F Sum_{k=0..n} x^k*T(n,k)= A009116(n), A000007(n), A011782(n), A006012(n), A083881(n), A081335(n), A090139(n), A145301(n), A145302(n), A145303(n), A143079(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - _Philippe Deléham_, Dec 04 2006, Oct 15 2008, Oct 19 2008

%F T(n,k) = T(n-1,k-1) + Sum_{i=0..k-1} T(n-2-i,k-1-i); T(0,0)=1; T(n,k)=0 if n < 0 or k < 0 or n < k. E.g.: T(8,5) = T(7,4) + T(6,4) + T(5,3) + T(4,2) + T(3,1) + T(2,0) = 7+15+5+1+0+0 = 28. - _Philippe Deléham_, Dec 04 2006

%F Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 respectively. - _Philippe Deléham_, Dec 24 2007

%F Sum_{k=0..n} T(n,k)*(-x)^(n-k) = A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x = 0,1,2,3,4,5 respectively. - _Philippe Deléham_, Nov 14 2008

%F T(n,k) = A085478(k,n-k). - _Philippe Deléham_, Dec 02 2008

%F T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1) - T(n-2,k-2), T(0,0) = T(1,1) = 1, T(1,0) = 0 and T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Mar 15 2012

%e Rows begin

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 0, 3, 1;

%e 0, 0, 1, 6, 1;

%t Table[Binomial[n, 2*(n-k)], {n,0,12}, {k,0,n}]//Flatten (* _Michael De Vlieger_, Oct 12 2016 *)

%o (PARI) {T(n,k)=polcoeff(polcoeff((1-x*y)/((1-x*y)^2-x^2*y)+x*O(x^n), n, x) + y*O(y^k),k,y)} (Hanna)

%o (PARI) T(n,k) = binomial(n, 2*(n-k));

%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Aug 01 2019

%o (Magma) [Binomial(n, 2*(n-k)): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 01 2019

%o (Sage) [[binomial(n, 2*(n-k)) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Aug 01 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(n, 2*(n-k)) ))); # _G. C. Greubel_, Aug 01 2019

%Y Cf. A098157, A034839.

%Y Cf. A119900. - _Philippe Deléham_, Dec 02 2008

%K easy,nonn,tabl

%O 0,9

%A _Paul Barry_, Aug 29 2004