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A094531 Array read by rows: right-hand side of triangle A027907 of trinomial coefficients. 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

LINKS

Table of n, a(n) for n=0..63.

Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.

He, Tian-Xiao, and Sprugnoli, Renzo; Sequence characterization of Riordan arrays, Discrete Math. 309 (2009), no. 12, 3962-3974.

A. Luzón, D. Merlini, M. A. Morón, R. 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.

P. Peart and W.-J. Woan, A divisibility property for a subgroup of Riordan matrices, Discrete Applied Mathematics, Vol. 98, Issue 3, Jan 2000, 255-263.

L. W. Shapiro, S. Getu, Wen-Jin Woan and L. 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.

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)

EXAMPLE

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;

  ...

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 *)

CROSSREFS

Binomial transform is triangle A094527. Row sums are A027914.

Cf. A111808 (row reversed).

Sequence in context: A115990 A350571 A277919 * A274293 A161009 A111960

Adjacent sequences:  A094528 A094529 A094530 * A094532 A094533 A094534

KEYWORD

easy,nonn,tabl

AUTHOR

Paul Barry, May 07 2004

STATUS

approved

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Last modified September 25 09:46 EDT 2022. Contains 356977 sequences. (Running on oeis4.)