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

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, 1, 393, 357, 266, 161, 77, 28, 7, 1, 1107, 1016, 784, 504, 266, 112, 36, 8, 1, 3139, 2907, 2304, 1554, 882, 414, 156, 45, 9, 1, 8953, 8350, 6765, 4740, 2850, 1452, 615, 210, 55

Offset: 0

Author

Paul Barry, May 07 2004

Keywords

• easy
• nonn
• tabl

Comments

Sometimes called a Motzkin triangle, although that name is usually reserved for A026300.

Expand (1+x+x^2)^n and take last (nonzero) coefficient of first row, last two coefficients of second row, etc.

Equals A094531*(1,xc(-x^2)) where c(x) is the g.f. of A000108. - Paul Barry, May 12 2009

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

Examples

			Triangle begins:
    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,   1;
  393, 357, 266, 161,  77,  28,   7,   1;
  ...
		

Links

• Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
• Tian-Xiao He and Renzo Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math. (2009) Vol 309, No. 12, 3962-3974.
• Ana Luzón, Donatella Merlini, Manuel A. Morón, and Renzo Sprugnoli, Complementary Riordan arrays, Discrete Applied Mathematics, 172 (2014) 75-87.
• Asamoah Nkwanta and Earl R. Barnes, Two Catalan-type Riordan Arrays and their Connections to the Chebyshev Polynomials of the First Kind, Journal of Integer Sequences, Article 12.3.3, 2012.
• Yassine Otmani, The 2-Pascal Triangle and a Related Riordan Array, J. Int. Seq. (2025) Vol. 28, Issue 3, Art. No. 25.3.5. See p. 23.
• Paul Peart and Wen-jin Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.
• Louis W. Shapiro, Seyoum Getu, Wen-Jin Woan, and Leon C. Woodson, The Riordan Group, Discrete Appl. Maths. 34 (1991) 229-239.
• Sheng-Liang Yang, Yan-Ni Dong, and Tian-Xiao He, Some matrix identities on colored Motzkin paths, Discrete Mathematics 340.12 (2017): 3081-3091.

Crossrefs

Binomial transform is triangle A094527. Row sums are A027914.

Cf. A111808 (row reversed).

Programs

Maple

T := (n, k) -> simplify(GegenbauerC(n-k, -n, -1/2)):
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, May 12 2016

Mathematica

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 *)
Table[Binomial[n, k] Hypergeometric2F1[(k - n)/2, (k - n + 1)/2, k + 1, 4], {n, 0, 9}, {k, 0, n}] // Flatten (* or *)
Table[If[n == 0, 1, GegenbauerC[n - k, -n, -1/2]], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 12 2016 *)

Formula

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

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

Column k has e.g.f. exp(x)*Bessel_I(k, 2x). - Paul Barry, Jul 14 2005

T(n, k) = Sum_{i=0..n} C(n-k-i, i)*C(n, k+i). - Paul Barry, Nov 04 2005

T(n, k) = Sum_{j=0..n} C(n,j)*C(j,n-k-j). - Paul Barry, Oct 25 2006

From Paul Barry, May 12 2009: (Start)

Production matrix is

1, 1;

2, 1, 1;

0, 1, 1, 1;

0, 0, 1, 1, 1;

0, 0, 0, 1, 1, 1; (End)

From Peter Bala, Jun 29 2015: (Start)

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).

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)

From Peter Luschny, May 12 2016: (Start)

T(n,k) = binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+1], 4):

T(n,k) = GegenbauerC(n-k, -n, -1/2). (End)