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%I A108267 #68 Apr 10 2024 11:03:17
%S A108267 1,1,1,1,7,1,1,31,31,1,1,121,381,121,1,1,456,3431,3431,456,1,1,1709,
%T A108267 26769,60691,26769,1709,1,1,6427,193705,848443,848443,193705,6427,1,1,
%U A108267 24301,1343521,10350421,19610233,10350421,1343521,24301,1
%N A108267 Triangle read by rows, T(n, k) = [x^k] (1-x)^(n+1)*Sum_{j=0..n} binomial(n + n*j + j, n*j + j)*x^j.
%C A108267 G.f. of row n divided by (1-x)^(n+1) equals g.f. of row n of table A060543.
%C A108267 Matrix product of this triangle with Pascal's triangle (A007318) equals A108291.
%C A108267 Seeing each row as a polynomial, all roots seem to be negative reals. - _F. Chapoton_, Nov 01 2022
%C A108267 From _Thomas Anton_, Jan 05 2023: (Start)
%C A108267 Consider the set [m] := {1, 2, 3, ..., m} ordered cyclically, and then mapped into itself via f. Let us consider a in [m] as the (a-1)th m-th root of unity e^(2*Pi*i*(a-1)/m). Then f may be extended to a continuous map f':S^1 -> S^1 as follows:
%C A108267 For a immediately before b in the cyclic order, map the interval between a and b to S^1 so that a point in it moving clockwise at constant speed has a value moving clockwise at constant speed, and the map travels the shortest distance possible given this condition.
%C A108267 T(n, k) gives the number of f for m = n-1 such that f(1) = 1 and f' has degree k. This is trivially one n-th of the number of f with degree k when f(1) is arbitrary.
%C A108267 Equivalent to having degree k is that there are k values a immediately before b in the cyclic order such that f(a) > f(b) (in the standard order of N).
%C A108267 If we change things so that a immediately before b satisfies f(a) = f(b) corresponds to a full rotation (this is equivalent to using the condition f(a) >= f(b) in the last paragraph), then T(n, k) is the number of f with degree k+1.
%C A108267 T(n, k) is the (k+1)*(n-1)th (n-1)-nomial coefficient of power n - 1.
%C A108267 (End)
%H A108267 M. Bayer, B. Goeckner, S. J. Hong, T. McAllister, M. Olsen, C. Pinckney, J. Vega and M. Yip, Lattice polytopes from Schur and symmetric Grothendieck polynomials, Electronic Journal of Combinatorics, Volume 28, Issue 2 (2021). See Proposition 53 and Table 1.
%H A108267 Tanay Wakhare, Iterated Entropy Derivatives and Binary Entropy Inequalities, arXiv:2312.14743 [cs.IT], 2023.
%H A108267 Tanay Wakhare, Two Studies of Constraints in High Dimensions: Entropy Inequalities and the Randomized Symmetric Binary Perceptron, Master's Thesis, MIT (2024). See p. 22.
%H A108267 Raphael Yuster, Almost k-union closed set systems, arXiv:2302.12276 [math.CO], 2023, p. 8.
%F A108267 T(n, 1) = A048775(n) = binomial(2*n + 1, n + 1) - (n + 1).
%F A108267 Sum_{k=0..n} T(n, k) = A000169(n) = (n + 1)^n.
%F A108267 Sum_{k=0..n} T(n, k)*2^k = A108292(n).
%F A108267 From _Thomas Anton_, Jan 05 2023: (Start)
%F A108267 T(n, k) = Sum_{i=0..k} (-1)^i*binomial(n + 1, i)*binomial(n+(n+1)*(k-i), n).
%F A108267 T(n, k) = T(n, n-k).
%F A108267 (End)
%e A108267 Triangle begins:
%e A108267 1;
%e A108267 1, 1;
%e A108267 1, 7, 1;
%e A108267 1, 31, 31, 1;
%e A108267 1, 121, 381, 121, 1;
%e A108267 1, 456, 3431, 3431, 456, 1;
%e A108267 1, 1709, 26769, 60691, 26769, 1709, 1;
%e A108267 1, 6427, 193705, 848443, 848443, 193705, 6427, 1;
%e A108267 ...
%e A108267 G.f. of row 3: (1 + 31*x + 31*x^2 + x^3) = (1-x)^4*(1 + 35*x + 165*x^2 + 455*x^3 + ... + C(4*j+3,4*j)*x^j + ...).
%p A108267 p := n -> (1-x)^(n+1)*add(binomial(n + n*j + j, n*j + j)*x^j, j = 0..n):
%p A108267 seq(print(seq(coeff(p(n), x, k), k = 0..n)), n = 0..8); # _Peter Luschny_, Nov 02 2022
%t A108267 T[n_, k_] := Coefficient[(1 - x)^(n + 1)*
%t A108267 Sum[Binomial[n + n*j + j, n*j + j]*x^j, {j, 0, n}], x, k];
%t A108267 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 23 2021 *)
%o A108267 (PARI) T(n,k)=polcoeff((1-x)^(n+1)*sum(j=0,n,binomial(n+n*j+j,n*j+j)*x^j),k)
%Y A108267 Cf. A108267, A000169, A048775, A060543, A108291, A108292, A056856.
%K A108267 nonn,tabl
%O A108267 0,5
%A A108267 _Paul D. Hanna_, May 29 2005 and May 31 2005
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