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A227532 Logarithmic derivative, wrt x, of triangle A227543, as read by terms k=0..n*(n-1)/2 in rows n>=1.

%I A227532 #35 Sep 10 2021 21:58:55
%S A227532 1,1,2,1,3,3,3,1,4,6,8,8,4,4,1,5,10,15,20,20,20,15,10,5,5,1,6,15,26,
%T A227532 39,48,57,60,54,48,36,30,18,12,6,6,1,7,21,42,70,98,126,154,168,175,
%U A227532 168,154,133,112,84,70,49,35,21,14,7,7,1,8,28,64,118,184,256,336,408,472,516,536,532,504,464,408,360,296,248,192,152,112,88,56,40,24,16,8,8
%N A227532 Logarithmic derivative, wrt x, of triangle A227543, as read by terms k=0..n*(n-1)/2 in rows n>=1.
%H A227532 Paul D. Hanna, <a href="/A227532/b227532.txt">Table of n, a(n) for n = 1..1350 (rows n=1..20 of triangle, flattened).</a>
%H A227532 Stéphane Ouvry and Alexios P. Polychronakos, <a href="https://arxiv.org/abs/2105.14042">Exclusion statistics for particles with a discrete spectrum</a>, arXiv:2105.14042 [cond-mat.stat-mech], 2021.
%F A227532 L.g.f.: Sum_{k=0..n*(n-1)/2, n>=1} T(n,k)*x^n*q^k/n = Log(G(x,q)) where G(x,q) = 1 + x*G(q*x,q)*G(x,q) is the g.f. of triangle A227543.
%F A227532 Row sums form A001700, the logarithmic derivative of the Catalan numbers.
%F A227532 Sum_{k=0..n*(n-1)/2} T(n,k) = binomial(2*n-1, n-1), for n>=1.
%F A227532 Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = (-1)^[n/2] * binomial(n-1, [(n-1)/2]).
%F A227532 Sum_{k=0..n*(n-1)/2} k*T(n,k) = n*2^(2*n-2) - (2*n-1)*binomial(2*n-2,n-1) = A153338(n), for n>=1.
%F A227532 Sum_{k=0..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n) = (-1)^(n-1) for n>=1; i.e., the n-th row sum at q = exp(2Pi*I/n), the n-th root of unity, equals -(-1)^n for n>=1.
%F A227532 Sum_{k=0..[n/2]} T(n, n*k) = A145855(n), the number of n-member subsets of 1..2n-1 whose elements sum to a multiple of n.
%F A227532 L.g.f. satisfies: L'(x,q) = P'(x,q)/P(x,q) - Q'(x,q)/Q(x,q), where
%F A227532   P(x,q) = Sum_{n>=0} q^(n^2) * (-x)^n / Product_{k=1..n} (1-q^k),
%F A227532   Q(x,q) = Sum_{n>=0} q^(n*(n-1)) * (-x)^n / Product_{k=1..n} (1-q^k),
%F A227532 due to Ramanujan's continued fraction identity. - _Paul D. Hanna_, Dec 28 2016
%e A227532 L.g.f.: L(x,q) = x*(1) + x^2*(1 + 2*q)/2 + x^3*(1 + 3*q + 3*q^2 + 3*q^3)/3
%e A227532 + x^4*(1 + 4*q + 6*q^2 + 8*q^3 + 8*q^4 + 4*q^5 + 4*q^6)/4
%e A227532 + x^5*(1 + 5*q + 10*q^2 + 15*q^3 + 20*q^4 + 20*q^5 + 20*q^6 + 15*q^7 + 10*q^8 + 5*q^9 + 5*q^10)/5
%e A227532 + x^6*(1 + 6*q + 15*q^2 + 26*q^3 + 39*q^4 + 48*q^5 + 57*q^6 + 60*q^7 + 54*q^8 + 48*q^9 + 36*q^10 + 30*q^11 + 18*q^12 + 12*q^13 + 6*q^14 + 6*q^15)/6 +...
%e A227532 where exponentiation yields the g.f. of triangle A227543:
%e A227532 exp(L(x,q)) = 1 + x*(1) + x^2*(1 + q) + x^3*(1 + 2*q + q^2 + q^3)
%e A227532 + x^4*(1 + 3*q + 3*q^2 + 3*q^3 + 2*q^4 + q^5 + q^6)
%e A227532 + x^5*(1 + 4*q + 6*q^2 + 7*q^3 + 7*q^4 + 5*q^5 + 5*q^6 + 3*q^7 + 2*q^8 + q^9 + q^10)
%e A227532 + x^6*(1 + 5*q + 10*q^2 + 14*q^3 + 17*q^4 + 16*q^5 + 16*q^6 + 14*q^7 + 11*q^8 + 9*q^9 + 7*q^10 + 5*q^11 + 3*q^12 + 2*q^13 + q^14 + q^15) +...
%e A227532 This triangle begins:
%e A227532 1;
%e A227532 1, 2;
%e A227532 1, 3, 3, 3;
%e A227532 1, 4, 6, 8, 8, 4, 4;
%e A227532 1, 5, 10, 15, 20, 20, 20, 15, 10, 5, 5;
%e A227532 1, 6, 15, 26, 39, 48, 57, 60, 54, 48, 36, 30, 18, 12, 6, 6;
%e A227532 1, 7, 21, 42, 70, 98, 126, 154, 168, 175, 168, 154, 133, 112, 84, 70, 49, 35, 21, 14, 7, 7;
%e A227532 1, 8, 28, 64, 118, 184, 256, 336, 408, 472, 516, 536, 532, 504, 464, 408, 360, 296, 248, 192, 152, 112, 88, 56, 40, 24, 16, 8, 8;
%e A227532 1, 9, 36, 93, 189, 324, 489, 684, 891, 1101, 1305, 1476, 1611, 1683, 1701, 1665, 1593, 1476, 1350, 1197, 1053, 900, 765, 630, 522, 405, 324, 243, 189, 135, 99, 63, 45, 27, 18, 9, 9; ...
%o A227532 (PARI) {T(n, k)=local(A=1); for(i=1, n, A=1+x*subst(A, x, q*x)*A +x*O(x^n)); n*polcoeff(polcoeff(log(A), n, x), k, q)}
%o A227532 for(n=1, 10, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print(""))
%o A227532 (PARI) /* By Ramanujan's continued fraction identity: */
%o A227532 {T(n, k)=local(P=1, Q=1);
%o A227532 P=sum(m=0, n+1, q^(m^2)*(-x)^m/prod(k=1, m, 1-q^k) +O(x^(n+2)));
%o A227532 Q=sum(m=0, n+1, q^(m*(m-1))*(-x)^m/prod(k=1, m, 1-q^k) +O(x^(n+2)));
%o A227532 polcoeff(polcoeff(P'/P - Q'/Q, n-1, x), k, q)}
%o A227532 for(n=1, 10, for(k=0, n*(n-1)/2, print1(T(n, k), ", ")); print("")) \\ _Paul D. Hanna_, Dec 28 2016
%Y A227532 Cf. A227543, A145855, A001700, A153338.
%K A227532 nonn
%O A227532 1,3
%A A227532 _Paul D. Hanna_, Jul 14 2013