A089942 Inverse binomial matrix applied to A039599.

1, 0, 1, 1, 1, 1, 1, 3, 2, 1, 3, 6, 6, 3, 1, 6, 15, 15, 10, 4, 1, 15, 36, 40, 29, 15, 5, 1, 36, 91, 105, 84, 49, 21, 6, 1, 91, 232, 280, 238, 154, 76, 28, 7, 1, 232, 603, 750, 672, 468, 258, 111, 36, 8, 1, 603, 1585, 2025, 1890, 1398, 837, 405, 155, 45, 9, 1, 1585, 4213, 5500

Offset

0,8

Comments

Reverse of A071947 - related to lattice paths. First column is A005043.

Triangle T(n,k), 0 <= k <= n, defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) for k >= 1. - Philippe Deléham, Feb 27 2007

This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - Philippe Deléham, Sep 25 2007

Riordan array (f(x),x*g(x)), where f(x)is the o.g.f. of A005043 and g(x)is the o.g.f. of A001006. - Philippe Deléham, Nov 22 2009

Riordan array ((1+x-sqrt(1-2x-3x^2))/(2x(1+x)), (1-x-sqrt(1-2x-3x^2))/(2x)). Inverse of Riordan array ((1+x)/(1+x+x^2),x/(1+x+x^2)). E.g.f. of column k is exp(x)*(Bessel_I(k,2x)-Bessel_I(k+1,2x)).

Diagonal sums are A187306.

Simultaneous equations using the first n rows solve for diagonal lengths of odd N = (2n+1) regular polygons, with constants c^0, c^1, c^2, ...; where c = 1 + 2*cos( 2*Pi/N) = sin(3*Pi/N)/sin(Pi/N) = the third longest diagonal of N>5. By way of example, take the first 4 rows relating to the 9-gon (nonagon), N=(2*4 + 1), with c = 1 + 2*cos(2*Pi/9) = 2.5320888.... The simultaneous equations are (1,0,0,0) = 1; (0,1,0,0) = c; (1,1,1,0) = c^2, (1,3,2,1) = c^3. The answers are 1, 2.532..., 2.879..., and 1.879...; the four distinct diagonal lengths of the 9-gon (nonagon) with edge = 1. - Gary W. Adamson, Sep 07 2011

Links

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

P. Barry and A. Hennessy, Four-term Recurrences, Orthogonal Polynomials and Riordan Arrays, Journal of Integer Sequences, 2012, article 12.4.2. - From N. J. A. Sloane, Sep 21 2012

E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Applied Mathematics, 34 (2005) pp. 101-122.

D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Sun, Yidong; Ma, Luping Minors of a class of Riordan arrays related to weighted partial Motzkin paths. Eur. J. Comb. 39, 157-169 (2014) Table 2.2

Formula

G.f.: (1+z-q)/[(1+z)(2z-t+tz+tq)], where q = sqrt(1-2z-3z^2).

Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A005043(m+n). - Philippe Deléham, Mar 22 2007

Sum_{k=0..n} T(n,k)*(2k+1) = 3^n. - Philippe Deléham, Mar 22 2007

Sum_{k=0..n} T(n,k)*2^k = A112657(n). - Philippe Deléham, Apr 01 2007

T(n,2k) + T(n,2k+1) = A109195(n,k). - Philippe Deléham, Nov 11 2008

T(n,k) = GegenbauerC(n-k,-n+1,-1/2) - GegenbauerC(n-k-1,-n+1,-1/2) for 1 <= k <= n. - Peter Luschny, May 12 2016

Example

Triangle begins
   1,
   0,   1,
   1,   1,   1,
   1,   3,   2,   1,
   3,   6,   6,   3,   1,
   6,  15,  15,  10,   4,  1,
  15,  36,  40,  29,  15,  5,  1,
  36,  91, 105,  84,  49, 21,  6, 1,
  91, 232, 280, 238, 154, 76, 28, 7, 1
Production matrix is
  0, 1,
  1, 1, 1,
  0, 1, 1, 1,
  0, 0, 1, 1, 1,
  0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 1, 1, 1,
  0, 0, 0, 0, 0, 0, 0, 1, 1, 1

Maple

T:= (n, k) -> simplify(GegenbauerC(n-k, -n+1, -1/2)-GegenbauerC(n-k-1, -n+1, -1/2)): for n from 1 to 9 do seq(T(n, k), k=1..n) od; # Peter Luschny, May 12 2016
# Or by recurrence:
T := proc(n, k) option remember;
if n = k then 1 elif k < 0 or n < 0 or k > n then 0
elif k = 0 then T(n-1, 1) else T(n-1, k-1) + T(n-1, k) + T(n-1, k+1) fi end:
for n from 0 to 9 do seq(T(n, k), k = 0..n) od; # Peter Luschny, May 25 2021

Mathematica

T[n_, k_] := GegenbauerC[n - k, -n + 1, -1/2] - GegenbauerC[n - k - 1, -n + 1, -1/2]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)

Crossrefs

Row sums give A002426 (central trinomial coefficients).

Sequence in context: A335012 A158275 A147750 * A097409 A257556 A078268

Adjacent sequences: A089939 A089940 A089941 * A089943 A089944 A089945

Keyword

nonn,tabl

Author

Paul Barry, Nov 16 2003

Extensions

Edited by Emeric Deutsch, Mar 04 2004

Status

approved