A291844 - OEIS

%I #28 Nov 17 2018 15:12:28

%S 1,1,4,2,29,23,274,292,36,3145,4068,994,16,42294,62861,22250,1512,

%T 651227,1075562,484840,61027,1060,11295242,20275944,10867381,1977879,

%U 93188,280,217954807,418724047,255929070,59896915,4823178,80632,4632600152,9418874022,6387031115,1798212190,204846125,7410676,37056,107572674851,229535650138,169414005231,55017177704,8022471066,463514918,7255380,7040

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

%C Row n>0 contains floor((2*n+2)/3) terms.

%H Gheorghe Coserea, <a href="/A291844/b291844.txt">Rows n = 0..124, flattened</a>

%H 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 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 A294158(n) = P_n(1), A294159(n)=P_n(-1), A294160(n)=P_n(0).

%e A(x;t) = 1 + x + (4 + 2*t)*x^2 + (29 + 23*t)*x^3 + (274 + 292*t + 36*t^2)*x^4 + ...

%e Triangle starts:

%e n\k  [0]        [1]        [2]        [3]       [4]      [5]

%e [0]  1;

%e [1]  1;

%e [2]  4,         2;

%e [3]  29,        23;

%e [4]  274,       292,       36;

%e [5]  3145,      4068,      994,       16;

%e [6]  42294,     62861,     22250,     1512;

%e [7]  651227,    1075562,   484840,    61027,    1060;

%e [8]  11295242,  20275944,  10867381,  1977879,  93188,   280;

%e [9]  217954807, 418724047, 255929070, 59896915, 4823178, 80632;

%e [10] ...

%t m = maxExponent = 13; Z[_] = 0;

%t 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 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 CoefficientList[# + O[l]^m, l]& /@ Most @ CoefficientList[gamma[t], t] // Flatten (* _Jean-François Alcover_, Nov 17 2018 *)

%o (PARI)

%o A291843_ser(N, t='t) = {

%o   my(x='x+O('x^N), y=1, y1=0, n=1,

%o   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   while (n++,

%o    y1 = (2*x^2*y'*((-t^2 + t)*x + (-t + 1) + (t^2*x^2 + (t^2 + t)*x + t)*y) +

%o         (t*x^2 + t*x)*y^2 - (2*t^2*x^3 + 3*t*x^2 + (-t + 1)*x - 1))*dn;

%o    if (y1 == y, break); y = y1;); y;

%o };

%o A291844_ser(N, t='t) = {

%o   my(z = A291843_ser(N+1,t));

%o   ((1+x)*z - 1)*(1 + t*x)/((1-t + t*(1+x)*z)*x*z^2);

%o };

%o concat(apply(p->Vecrev(p), Vec(A291844_ser(12))))

%Y Cf. A286795, A286798, A286800, A291843.

%Y Columns k=0..5 give A294160 (k=0), A294161 (k=1), A294162 (k=2), A294163 (k=3), A294164 (k=4), A294165 (k=5).

%K nonn,tabf

%O 0,3

%A _Gheorghe Coserea_, Oct 24 2017