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A336110 Irregular triangle of Catalan-based numbers, read by rows.

%I A336110 #178 Dec 16 2024 14:54:48
%S A336110 1,1,2,-2,5,-14,5,14,-74,74,-14,42,-352,668,-352,42,132,-1588,4808,
%T A336110 -4808,1588,-132,429,-6946,30371,-48540,30371,-6946,429,1430,-29786,
%U A336110 176270,-407810,407810,-176270,29786,-1430
%N A336110 Irregular triangle of Catalan-based numbers, read by rows.
%C A336110 Calculation of the sum over the partitions of r of products of dimensions of two different representations of a symmetric group S_r gives
%C A336110 Sum_{L |- S_r} f(L)*f(l+q^N) = (r+q^N)! * G[N+1] * G[q+1]/(G[N+q+1]) * B_r(1!c_1, ..., r!c_r) where f(L) is the dimension of the symmetric group S_r, G[x] is Barnes function, and B_r() is the complete exponential Bell polynomial.
%C A336110 In the limit N -> infinity the coefficients [are?]
%C A336110   c_1 = 1/(1+x), c_i = 1/(i*N^(2*(i-1)))*P(i-1), for i >= 2.
%C A336110 Coefficient of x^n in the numerator of P(i) is T(s, i).
%C A336110 This triangle of coefficients was discovered by Borisenko et al. In mathematical physics these coefficients appear as an important ingredient of series that define the free energy of the SU(N) standard lattice model in the large N limit.
%C A336110 They are easily obtained from the g.f.
%C A336110 Some special cases are given by A000108 (first column of the triangle), A138156 (second column of the triangle).
%C A336110 The sum of the numbers in row 2*k+1 is (-1)^k * A000260(k) * 2^(2*k).
%H A336110 Sergii Voloshyn, <a href="/A336110/b336110.txt">Table of n, a(n) for n = 1..120</a>
%H A336110 O. Borisenko, V. Chelnokov, and Sergii Voloshyn, <a href="https://arxiv.org/abs/2008.00773">The large N limit of SU(N) integrals in lattice</a>, arXiv:2008.00773 [hep-lat], 2020. See formula (48).
%H A336110 F. Green and S. Samuel, <a href="https://doi.org/10.1016/0550-3213(82)90515-6">Calculating the large-N phase transition in gauge and matrix models</a>, Nuclear Physics B 194, Issue 1, 11 January 1982, Pages 107-156, See Appendix A.
%F A336110 Coefficients of x^n in the numerator of P(s) = (x * C[s]* 3F2[ s+ 1/2, s+1, s+3/2; 1/2,3/2; x^2] - x^2 * 4^s * 3F2[ s+1,s+3/2, s+2; 3/2, 2; x^2]), where C(s) are Catalan numbers.
%F A336110 or in a more explicit way (only for k >= 1)
%F A336110 T(s, n) = C(s) * U(s, n) - 4^s * V(s, n), where
%F A336110 U(s, n) = Sum_{a=0..(n-1)/2} (u(s, a)/u(0,a)) * M(s, n-1- 2a),
%F A336110 V(s, n) = Sum_{a=0..(n-2)/2} (v(s, a)/v(0,a)) * M(s, n-2- 2a), and
%F A336110 u(s, n) = Product_{L=1..n} binomial(2s+2L+1, 3),
%F A336110 v(s, n) = Product_{L=1..n} binomial(2s+2L+2, 3), and
%F A336110 M(m, n) = Sum_{L=1..n} L!/n! B_{n,l} ( x_1, ..., x_{n-L+1}), and
%F A336110 x_i = (-1)^{1+i} * (3s+i)_i = (-1)^{1+i} * i! * binomial(3s + i, i), where
%F A336110 B_{n,l} (x_1, ..., x_{n-L+1}) is the n-th partial or incomplete exponential Bell polynomial with monomials sorted into graded lexicographic order.
%F A336110 Sum of numbers in the particular row:
%F A336110 Sum_{n=1..2*k+1} T(2*k+1, n) = 2*(4*k+1)!/((k+1)!*(3*k+2)!) *2^(2*k) (odd s);
%F A336110 Sum_{n=1..2*k} T(2*k, n) = 0 (even s).
%F A336110 From _Sergii Voloshyn_, Oct 22 2020: (Start)
%F A336110 Formulae for particular columns:
%F A336110 T(s, 1) = C(s);
%F A336110 T(s, 2) = C(s)*(3*s+1) - 4^s;
%F A336110 T(s, 3) = C(s)*(binomial(2*s+3,3) + (3*s+1)^2 - binomial(3*s +2,2)) - 4^s*(3*s+1);
%F A336110 T(s, 4) = C(s)*((2*s+1)binomial(2*s+3,3) +(3*s+1)^3 - 2(3*s+1)* binomial(3*s+2,2)+ binomial(3*s+3, 3)) - 4^s*(binomial(3*s+4, 3)/4 + (3*s+1)^2 - binomial(3*s+2,2));
%F A336110 ...
%F A336110 T(s, s) = (-1)^(s+1)*C(s). (End)
%F A336110 From _Sergii Voloshyn_, Mar 17 2021: (Start)
%F A336110 Recursion ( P[0] = x/(1+x) ):
%F A336110 x*(d^3/dx^3)*P[s] = (1/4)*(2*k+2)*(2*k+3)*(2*k+4)*P[s+1]
%F A336110 for P[s_] := x CatalanNumber[s] HypergeometricPFQ[{s + 1/2, s + 1, s + 3/2}, {1/2, 3/2}, x^2] - x^2*4^s HypergeometricPFQ[{s + 1, s + 3/2, s + 2}, {3/2, 2}, x^2].
%F A336110 (End)
%F A336110 Recursion for array ( T(1,1) = 1 ): T(k, p) = (1/(k*(k+1)*(2k+1)))*[p*(p+1)*(p+2)*T(k-1,p+2) - 3*p*(p+1)*(3*k-p-1)*T(k-1,p+1) + 3*p*(3*k-p)*(3*k-p-1)*T(k-1,p) - (3*k-p+1)*(3*k-p)*(3*k-p-1)*T(k-1,p-1)]; p =[1,...,k]. - _Sergii Voloshyn_, Apr 14 2021
%F A336110 From _Sergii Voloshyn_, Apr 17 2021: (Start)
%F A336110 G.f.: Sum_{m>=1} x^m Om(k, m) = (1/(1+x)^(3*k+1))*Sum_{n=1..k} x^k * T(k,n);
%F A336110 Om(k, m) = (12^k/((1+k)!*(2k+1)!)) * Product_{L=1..k} binomial(m+2L, 3).
%F A336110 (End)
%F A336110 From _Sergii Voloshyn_, Apr 25 2021: (Start)
%F A336110 Differential equation for P[k_]:
%F A336110 x*(x^2-1)*(d^3/dx^3)P[s] + (2*k+2)*3*x^2*(d^2/dx^2)P[s] +(2*k+2)*(2*k+1)*3*x (d/dx)P[s] + (2*k+2)(2*k+1)*2*k*P[s] = 0.
%F A336110 Discrete set equation for T(k,n) (n=-1..k-2) at fixed k:
%F A336110 (k-n-1)*(k-n)*(k-n+1)*T(k,n) - (k-n-1)*(k-n)*(8*k+n+5)*T(k,n+1) - (n+1)*(n+2)*(9-n+3)*T(k,n+2) + (n+1)*(n+2)*(n+3)*T(k,n+3) = 0
%F A336110 and
%F A336110 Sum_{m=1..k} (k*(5+7*k) + 12*n*(n-1-k))*T(k,n) = 0. (End)
%F A336110 P[k_]:= (2^k)/((2*k+1)!*(k+1)!)*(-1/(1+x))^{2k+1}[Sum_{r(1)+...+ r(L)=k} (-x/(1+x))^{k-L+1}* 1^{(3*r(1))}*(1+3*r(1)-1)^{(3*r(2))}*... *(1+Sum_{i=1..L-1} 3 r(i) -L+1)^{(3*r(L))}] where Sum_ r_i = k runs over all integer compositions of k, L is a number of parts of this composition and 1^{(3 r(1))} is a rising factorial. - _Sergii Voloshyn_, Sep 03 2021
%F A336110 Sum_{k} abs(T(n,k)) = A000309(n-1). - _Sergii Voloshyn_, Nov 20 2024
%e A336110      1;
%e A336110      1;
%e A336110      2,     -2;
%e A336110      5,    -14,      5;
%e A336110     14,    -74,     74,     -14;
%e A336110     42,   -352,    668,    -352,     42;
%e A336110    132,  -1588,   4808,   -4808,   1588,    -132;
%e A336110    429,  -6946,  30371,  -48540,  30371,   -6946,   429;
%e A336110   1430, -29786, 176270, -407810, 407810, -176270, 29786, -1430;
%e A336110   ...
%t A336110 T[L_, n_] := CatalanNumber[L] Sum[u[L, a]/u[0, a] M[n - 1 - 2*a, L], {a, 0, (n - 1)/2}]-4^L Sum[v[L, a]/v[0, a] M[n - 2 - 2*a, L], {a,0,(n-2)/2}];
%t A336110 M[n_, l_] := Sum[k!/n! BellY[n,k,Table[(-1)^(j-1) j!Binomial[3l+j,j], {j,n}]], {k, 0, n}];
%t A336110 u[k_, n_] := Product[Binomial[2 k + 2 l + 1, 3], {l, 1, n}];
%t A336110 v[k_, n_] := Product[ Binomial[2 k + 2 l + 2, 3], {l, 1, n}];
%t A336110 (* alternate program using coefficients in numerator *)
%t A336110 P[s_] := x CatalanNumber[s] HypergeometricPFQ[{s + 1/2, s + 1, s + 3/2}, {1/2, 3/2}, x^2] - x^2*4^s HypergeometricPFQ[{s + 1, s + 3/2, s + 2}, {3/2, 2}, x^2];
%t A336110 Table[CoefficientList[P[s] // Together // Numerator, x] // Rest, {s, 0, 10}] // Flatten (* amended by _Jean-François Alcover_, Sep 25 2020 *)
%t A336110 (* another program using coefficients in numerator *)
%t A336110 Needs["Combinatorica`"];
%t A336110 OA[p_,x_]:= (2^p(-(1/(x+1)))^(2p+1))/((2p+1)!(p+1)!) Sum[(-x/(1+x))^(p-r+1)Product[Pochhammer[1+Plus@@Table[3*k[[i]]-1,{i,1,j-1}],3*k[[j]]],{j,1,r}],{r, 1, p},{k,Compositions[p-r,r]+1}]; Table[CoefficientList[OA[s, x] // Together // Numerator, x] //
%t A336110    Rest, {s, 0, 10}] // Flatten (* _Sergii Voloshyn_, Sep 03 2021 *)
%Y A336110 Cf. A000108, A000309, A138156, A000260.
%K A336110 sign,tabf
%O A336110 1,3
%A A336110 _Sergii Voloshyn_, Jul 08 2020