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A094531 Array read by rows: right-hand side of triangle A027907 of trinomial coefficients.

%I A094531 #60 May 23 2025 01:09:03
%S A094531 1,1,1,3,2,1,7,6,3,1,19,16,10,4,1,51,45,30,15,5,1,141,126,90,50,21,6,
%T A094531 1,393,357,266,161,77,28,7,1,1107,1016,784,504,266,112,36,8,1,3139,
%U A094531 2907,2304,1554,882,414,156,45,9,1,8953,8350,6765,4740,2850,1452,615,210,55
%N A094531 Array read by rows: right-hand side of triangle A027907 of trinomial coefficients.
%C A094531 Sometimes called a Motzkin triangle, although that name is usually reserved for A026300.
%C A094531 Expand (1+x+x^2)^n and take last (nonzero) coefficient of first row, last two coefficients of second row, etc.
%C A094531 Equals A094531*(1,xc(-x^2)) where c(x) is the g.f. of A000108. - _Paul Barry_, May 12 2009
%C A094531 Coefficients of Faber polynomials for (1/x+1+x): Fa(n,x) = Sum_{k=0..n} T(n,k)*x^k, g.f.: -log((sqrt(-3*t^2-2*t+1)-t+1)/2-t*x) = Sum_{n>0} Fa(n,x)*t^n/n. - _Vladimir Kruchinin_, Jul 01 2013
%H A094531 Paul Barry, <a href="http://dx.doi.org/10.1016/j.laa.2015.10.032">Riordan arrays, generalized Narayana triangles, and series reversion</a>, Linear Algebra and its Applications, 491 (2016) 343-385.
%H A094531 Tian-Xiao He and Renzo Sprugnoli, <a href="http://dx.doi.org/10.1016/j.disc.2008.11.021">Sequence characterization of Riordan arrays</a>, Discrete Math. (2009) Vol 309, No. 12, 3962-3974.
%H A094531 Ana Luzón, Donatella Merlini, Manuel A. Morón, and Renzo Sprugnoli, <a href="http://dx.doi.org/10.1016/j.dam.2014.03.005">Complementary Riordan arrays</a>, Discrete Applied Mathematics, 172 (2014) 75-87.
%H A094531 Asamoah Nkwanta and Earl R. Barnes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Nkwanta/nkwanta2.html">Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind</a>, Journal of Integer Sequences, Article 12.3.3, 2012.
%H A094531 Yassine Otmani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL28/Otmani/otmani10.html">The 2-Pascal Triangle and a Related Riordan Array</a>, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 23.
%H A094531 Paul Peart and Wen-jin Woan, <a href="http://dx.doi.org/10.1016/S0166-218X(99)00166-3">A divisibility property for a subgroup of Riordan matrices</a>, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
%H A094531 Louis W. Shapiro, Seyoum Getu, Wen-Jin Woan, and Leon C. Woodson, <a href="http://dx.doi.org/10.1016/0166-218X(91)90088-E">The Riordan Group</a>, Discrete Appl. Maths. 34 (1991) 229-239.
%H A094531 Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, <a href="http://dx.doi.org/10.1016/j.disc.2017.07.006">Some matrix identities on colored Motzkin paths</a>, Discrete Mathematics 340.12 (2017): 3081-3091.
%F A094531 Riordan array ( 1/sqrt(1-2*x-3*x^2), (1-x-sqrt(1-2*x-3*x^2))/(2*x) ). - _N. J. A. Sloane_, Jun 02 2005
%F A094531 Product of Riordan arrays (1/(1-x), x/(1-x)) (Pascal's triangle, A007318) and (1/sqrt(1-4x^2), (1-sqrt(1-4*x^2))/(2*x)) (A108044). Inverse is A102587. - _Paul Barry_, Jul 14 2005
%F A094531 Column k has e.g.f. exp(x)*Bessel_I(k, 2x). - _Paul Barry_, Jul 14 2005
%F A094531 T(n, k) = Sum_{i=0..n} C(n-k-i, i)*C(n, k+i). - _Paul Barry_, Nov 04 2005
%F A094531 T(n, k) = Sum_{j=0..n} C(n,j)*C(j,n-k-j). - _Paul Barry_, Oct 25 2006
%F A094531 From _Paul Barry_, May 12 2009: (Start)
%F A094531 Production matrix is
%F A094531   1, 1;
%F A094531   2, 1, 1;
%F A094531   0, 1, 1, 1;
%F A094531   0, 0, 1, 1, 1;
%F A094531   0, 0, 0, 1, 1, 1; (End)
%F A094531 From _Peter Bala_, Jun 29 2015: (Start)
%F A094531 Riordan array has the form ( x*h'(x)/h(x), h(x) ) with h(x) = (1 - x -sqrt(1 - 2*x - 3*x^2))/(2*x) and so belongs to the hitting time subgroup H of the Riordan group (see Peart and Woan, Example 1.1).
%F A094531 T(n,k) = [x^(n-k)] f(x)^n with f(x) = 1 + x + x^2. In general the (n,k)th entry of the hitting time array ( x*h'(x)/h(x), h(x) ) has the form [x^(n-k)] f(x)^n, where f(x) = x/( series reversion of h(x) ). (End)
%F A094531 From _Peter Luschny_, May 12 2016: (Start)
%F A094531 T(n,k) = binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+1], 4):
%F A094531 T(n,k) = GegenbauerC(n-k, -n, -1/2). (End)
%e A094531 Triangle begins:
%e A094531     1;
%e A094531     1,   1;
%e A094531     3,   2,   1;
%e A094531     7,   6,   3,   1;
%e A094531    19,  16,  10,   4,   1;
%e A094531    51,  45,  30,  15,   5,   1;
%e A094531   141, 126,  90,  50,  21,   6,   1;
%e A094531   393, 357, 266, 161,  77,  28,   7,   1;
%e A094531   ...
%p A094531 T := (n, k) -> simplify(GegenbauerC(n-k, -n, -1/2)):
%p A094531 for n from 0 to 9 do seq(T(n,k), k=0..n) od; # _Peter Luschny_, May 12 2016
%t A094531 max = 10; se = Series[ -Log[ (Sqrt[-3*t^2 - 2*t + 1] - t + 1)/2 - t*x], {t, 0, max + 1}, {x, 0, max}]; a[n_, k_] := SeriesCoefficient[se, {t, 0, n}, {x, 0, k}]*n; a[0, 0] = 1; Table[a[n, k], {n, 0, max }, {k, 0, n}] // Flatten  (* _Jean-François Alcover_, Jul 02 2013, after _Vladimir Kruchinin_ *)
%t A094531 Table[Binomial[n, k] Hypergeometric2F1[(k - n)/2, (k - n + 1)/2, k + 1, 4], {n, 0, 9}, {k, 0, n}] // Flatten (* or *)
%t A094531 Table[If[n == 0, 1, GegenbauerC[n - k, -n, -1/2]], {n, 0, 9}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, May 12 2016 *)
%Y A094531 Binomial transform is triangle A094527. Row sums are A027914.
%Y A094531 Cf. A111808 (row reversed).
%K A094531 easy,nonn,tabl
%O A094531 0,4
%A A094531 _Paul Barry_, May 07 2004