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A291844 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A291844 #28 Nov 17 2018 15:12:28
%S A291844 1,1,4,2,29,23,274,292,36,3145,4068,994,16,42294,62861,22250,1512,
%T A291844 651227,1075562,484840,61027,1060,11295242,20275944,10867381,1977879,
%U A291844 93188,280,217954807,418724047,255929070,59896915,4823178,80632,4632600152,9418874022,6387031115,1798212190,204846125,7410676,37056,107572674851,229535650138,169414005231,55017177704,8022471066,463514918,7255380,7040
%N A291844 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%C A291844 Row n>0 contains floor((2*n+2)/3) terms.
%H A291844 Gheorghe Coserea, <a href="/A291844/b291844.txt">Rows n = 0..124, flattened</a>
%H A291844 Luca G. Molinari, Nicola Manini, <a href="https://arxiv.org/abs/cond-mat/0512342">Enumeration of many-body skeleton diagrams</a>, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006.
%F A291844 y(x;t) = Sum_{n>=0} P_n(t)*x^n satisfies y = ((1+x)*z - 1) * (1 + t*x)/((1-t + t*(1+x)*z)*x*z^2), where z = A291843(x;t) and P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n > 0.
%F A291844 A294158(n) = P_n(1), A294159(n)=P_n(-1), A294160(n)=P_n(0).
%e A291844 A(x;t) = 1 + x + (4 + 2*t)*x^2 + (29 + 23*t)*x^3 + (274 + 292*t + 36*t^2)*x^4 + ...
%e A291844 Triangle starts:
%e A291844 n\k  [0]        [1]        [2]        [3]       [4]      [5]
%e A291844 [0]  1;
%e A291844 [1]  1;
%e A291844 [2]  4,         2;
%e A291844 [3]  29,        23;
%e A291844 [4]  274,       292,       36;
%e A291844 [5]  3145,      4068,      994,       16;
%e A291844 [6]  42294,     62861,     22250,     1512;
%e A291844 [7]  651227,    1075562,   484840,    61027,    1060;
%e A291844 [8]  11295242,  20275944,  10867381,  1977879,  93188,   280;
%e A291844 [9]  217954807, 418724047, 255929070, 59896915, 4823178, 80632;
%e A291844 [10] ...
%t A291844 m = maxExponent = 13; Z[_] = 0;
%t A291844 Do[Z[t_] = -(((1 - l + l (1+t) Z[t]) (-((t Z[t])/(1 + l t)) - (1 - t - 2 l t^2)/(1 - l + l (1+t) Z[t]) - 2 t^2 Z'[t]))/((1+t) (1 - t - 2 l t^2))) + O[t]^m // Normal // Simplify, {m}];
%t A291844 gamma[t_] = ((1 + l t)(-1 + Z[t] + t Z[t]))/(Z[t]^2 (t + l t (-1 + Z[t] + t Z[t]))) + O[t]^m // Normal // Simplify;
%t A291844 CoefficientList[# + O[l]^m, l]& /@ Most @ CoefficientList[gamma[t], t] // Flatten (* _Jean-François Alcover_, Nov 17 2018 *)
%o A291844 (PARI)
%o A291844 A291843_ser(N, t='t) = {
%o A291844   my(x='x+O('x^N), y=1, y1=0, n=1,
%o A291844   dn = 1/(-2*t^2*x^4 - (2*t^2+3*t)*x^3 - (2*t+1)*x^2 + (2*t-1)*x + 1));
%o A291844   while (n++,
%o A291844    y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +
%o A291844         (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;
%o A291844    if (y1 == y, break); y = y1;); y;
%o A291844 };
%o A291844 A291844_ser(N, t='t) = {
%o A291844   my(z = A291843_ser(N+1,t));
%o A291844   ((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);
%o A291844 };
%o A291844 concat(apply(p->Vecrev(p), Vec(A291844_ser(12))))
%Y A291844 Cf. A286795, A286798, A286800, A291843.
%Y A291844 Columns k=0..5 give A294160 (k=0), A294161 (k=1), A294162 (k=2), A294163 (k=3), A294164 (k=4), A294165 (k=5).
%K A291844 nonn,tabf
%O A291844 0,3
%A A291844 _Gheorghe Coserea_, Oct 24 2017