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A278457 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A278457 #23 Sep 28 2019 12:30:02
%S A278457 1,2,2,7,11,6,30,65,59,22,143,397,492,318,90,728,2471,3857,3430,1728,
%T A278457 394,3876,15572,29255,32728,22886,9461,1806,21318,99009,217323,291456,
%U A278457 257001,148626,52133,8558,120175,633765,1591231,2481597,2622445,1918665,947740,288812,41586,690690,4078360,11527318,20467755,25114375,22043890,13821764,5964728,1607198,206098
%N A278457 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%H A278457 Gheorghe Coserea, <a href="/A278457/b278457.txt">Rows n = 1..101, flattened</a>
%H A278457 F. Chapoton, F. Hivert, J.-C. Novelli, <a href="http://arxiv.org/abs/1307.0092">A set-operad of formal fractions and dendriform-like sub-operads</a>, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
%F A278457 y(x) = Sum {n>=1} P_n(t)*x^n satisfies x = y*((t-1)*y^3 + (t^2-2*t-1)*y^2 - (2*t-1)*y + 1)/((1-t)*y^3 + (3-t)*y^2 + 3*y + 1), with y(0)=0, y'(0)=1, where P_n(t) is the degree n-1 polynomial associated with row n of the triangle in order of decreasing powers of t.
%F A278457 P_n(0) = A006318(n-1), P_n(1) = A156017(n-1), P_n(2) = A231690(n).
%e A278457 A(x;t) = x + (2*t+2)*x^2 + (7*t^2+11*t+6)*x^3 + (30*t^3+65*t^2+59*t+22)*x^4 + ...
%e A278457 Triangle starts:
%e A278457 n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]      [8]
%e A278457 [1]  1;
%e A278457 [2]  2,       2;
%e A278457 [3]  7,       11,      6;
%e A278457 [4]  30,      65,      59,      22;
%e A278457 [5]  143,     397,     492,     318,     90;
%e A278457 [6]  728,     2471,    3857,    3430,    1728,    394;
%e A278457 [7]  3876,    15572,   29255,   32728,   22886,   9461,    1806;
%e A278457 [8]  21318,   99009,   217323,  291456,  257001,  148626,  52133,   8558;
%e A278457 [9]  ...
%t A278457 Reverse[CoefficientList[#, t]]& /@ CoefficientList[InverseSeries[x ((t - 1) x^3 + (t^2 - 2t - 1) x^2 - (2t - 1) x + 1)/((1 - t) x^3 + (3 - t) x^2 + 3x + 1) + O[x]^11], x] // Flatten (* _Jean-François Alcover_, Sep 28 2019 *)
%o A278457 (PARI)
%o A278457 N=11; x ='x + O('x^N);
%o A278457 concat(apply(p->Vec(p), Vec(serreverse(Ser(x*((t-1)*x^3 + (t^2-2*t-1)*x^2 - (2*t-1)*x+1)/((1-t)*x^3 + (3-t)*x^2 + 3*x + 1), 'x)))))
%Y A278457 Column k=1 give A006013.
%K A278457 nonn,tabl
%O A278457 1,2
%A A278457 _Gheorghe Coserea_, Jan 15 2017