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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)).

%I A185967 #24 Apr 11 2024 10:50:11
%S A185967 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,
%T A185967 25,6,1,2563,1818,897,359,120,33,7,1,11181,7946,3966,1628,570,170,42,
%U A185967 8,1,49720,35389,17823,7458,2701,852,231,52,9,1,224540,160024,81177,34484,12815,4212,1218,304,63,10,1
%N 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)).
%C A185967 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.
%H A185967 Emeric Deutsch, Luca Ferrari, and Simone Rinaldi, <a href="https://doi.org/10.1016/j.aam.2004.05.002">Production Matrices</a>, Advances in Mathematics, 34 (2005) pp. 101-122.
%H A185967 JiSun Huh, Sangwook Kim, Seunghyun Seo, and Heesung Shin, <a href="https://arxiv.org/abs/2404.04091">Bijections on pattern avoiding inversion sequences and related objects</a>, arXiv:2404.04091 [math.CO], 2024. See p. 22.
%F A185967 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
%F A185967 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
%e A185967 Triangle begins
%e A185967   1,
%e A185967   1, 1,
%e A185967   3, 2, 1,
%e A185967   10, 7, 3, 1,
%e A185967   37, 26, 12, 4, 1,
%e A185967   146, 103, 49, 18, 5, 1,
%e A185967   602, 426, 207, 80, 25, 6, 1,
%e A185967   2563, 1818, 897, 359, 120, 33, 7, 1,
%e A185967   11181, 7946, 3966, 1628, 570, 170, 42, 8, 1,
%e A185967   49720, 35389, 17823, 7458, 2701, 852, 231, 52, 9, 1,
%e A185967   224540, 160024, 81177, 34484, 12815, 4212, 1218, 304, 63, 10, 1
%e A185967 Production matrix is
%e A185967   1, 1,
%e A185967   2, 1, 1,
%e A185967   3, 2, 1, 1,
%e A185967   4, 3, 2, 1, 1,
%e A185967   5, 4, 3, 2, 1, 1,
%e A185967   6, 5, 4, 3, 2, 1, 1,
%e A185967   7, 6, 5, 4, 3, 2, 1, 1,
%e A185967   8, 7, 6, 5, 4, 3, 2, 1, 1,
%e A185967   9, 8, 7, 6, 5, 4, 3, 2, 1, 1
%e A185967   10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1
%p A185967 T := (n, k) -> `if`(n=k, 1, (1 + k)*(n - k)*hypergeom([1 + k - n, -n, 1 - k + n], [3/2, 2], 1/4)):
%p A185967 seq(seq(simplify(T(n, k)), k=0..n),n=0..10); # _Peter Luschny_, Apr 02 2019
%t A185967 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_ *)
%o A185967 (Maxima)
%o A185967 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 */
%Y A185967 Inverse of number triangle A185962.
%Y A185967 First column is A109081. Row sums are A106228(n+1).
%K A185967 nonn,easy,tabl
%O A185967 0,4
%A A185967 _Paul Barry_, Feb 07 2011