A185967 Inverse of Riordan array ((1-x)(1-x^2)(1-x^3)/(1-x^6), x(1-x)(1-x^2)(1-x^3)/(1-x^6)).

1, 1, 1, 3, 2, 1, 10, 7, 3, 1, 37, 26, 12, 4, 1, 146, 103, 49, 18, 5, 1, 602, 426, 207, 80, 25, 6, 1, 2563, 1818, 897, 359, 120, 33, 7, 1, 11181, 7946, 3966, 1628, 570, 170, 42, 8, 1, 49720, 35389, 17823, 7458, 2701, 852, 231, 52, 9, 1, 224540, 160024, 81177, 34484, 12815, 4212, 1218, 304, 63, 10, 1

Offset

0,4

Comments

Riordan array (g(x),xg(x)) where x*g(x) = (x+2)/3 - 2*sqrt(1+x+x^2) * cos(arccos(-(2x^3+3x^2+24x-2) / (2(1+x+x^2)^(3/2)))/3)/3.

Links

Table of n, a(n) for n=0..65.

Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.

JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, Bijections on pattern avoiding inversion sequences and related objects, arXiv:2404.04091 [math.CO], 2024. See p. 22.

Formula

T(n, k) = (k + 1)/(n + 1)*Sum_{i=0..n-k} C(n+1, i)*C(n-k+i-1, n-k-i). - Vladimir Kruchinin, Apr 02 2019

T(n, k) = (1 + k)*(n - k)*hypergeom([1 + k - n, -n, 1 - k + n], [3/2, 2], 1/4) for k < n. - Peter Luschny, Apr 02 2019

Example

Triangle begins
  1,
  1, 1,
  3, 2, 1,
  10, 7, 3, 1,
  37, 26, 12, 4, 1,
  146, 103, 49, 18, 5, 1,
  602, 426, 207, 80, 25, 6, 1,
  2563, 1818, 897, 359, 120, 33, 7, 1,
  11181, 7946, 3966, 1628, 570, 170, 42, 8, 1,
  49720, 35389, 17823, 7458, 2701, 852, 231, 52, 9, 1,
  224540, 160024, 81177, 34484, 12815, 4212, 1218, 304, 63, 10, 1
Production matrix is
  1, 1,
  2, 1, 1,
  3, 2, 1, 1,
  4, 3, 2, 1, 1,
  5, 4, 3, 2, 1, 1,
  6, 5, 4, 3, 2, 1, 1,
  7, 6, 5, 4, 3, 2, 1, 1,
  8, 7, 6, 5, 4, 3, 2, 1, 1,
  9, 8, 7, 6, 5, 4, 3, 2, 1, 1
  10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1

Maple

T := (n, k) -> `if`(n=k, 1, (1 + k)*(n - k)*hypergeom([1 + k - n, -n, 1 - k + n], [3/2, 2], 1/4)):
seq(seq(simplify(T(n, k)), k=0..n), n=0..10); # Peter Luschny, Apr 02 2019

Mathematica

T[n_, k_] := (k+1)/(n+1) Sum[Binomial[n+1, i] Binomial[n-k+i-1, n-k-i], {i, 0, n-k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 25 2019, after Vladimir Kruchinin *)

Prog

(Maxima)
T(n, k):=(k+1)/(n+1)*sum(binomial(n+1, i)*binomial(n-k+i-1, n-k-i), i, 0, n-k); /* Vladimir Kruchinin, Apr 02 2019 */

Crossrefs

Inverse of number triangle A185962.

First column is A109081. Row sums are A106228(n+1).

Sequence in context: A100100 A248036 A307214 * A188111 A102472 A267629

Adjacent sequences: A185964 A185965 A185966 * A185968 A185969 A185970

Keyword

nonn,easy,tabl

Author

Paul Barry, Feb 07 2011

Status

approved