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

1, 2, 2, 7, 11, 6, 30, 65, 59, 22, 143, 397, 492, 318, 90, 728, 2471, 3857, 3430, 1728, 394, 3876, 15572, 29255, 32728, 22886, 9461, 1806, 21318, 99009, 217323, 291456, 257001, 148626, 52133, 8558, 120175, 633765, 1591231, 2481597, 2622445, 1918665, 947740, 288812, 41586, 690690, 4078360, 11527318, 20467755, 25114375, 22043890, 13821764, 5964728, 1607198, 206098

Offset: 1

Author

Gheorghe Coserea, Jan 15 2017

Keywords

• nonn
• tabl

Examples

			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 + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]      [8]
[1]  1;
[2]  2,       2;
[3]  7,       11,      6;
[4]  30,      65,      59,      22;
[5]  143,     397,     492,     318,     90;
[6]  728,     2471,    3857,    3430,    1728,    394;
[7]  3876,    15572,   29255,   32728,   22886,   9461,    1806;
[8]  21318,   99009,   217323,  291456,  257001,  148626,  52133,   8558;
[9]  ...
		

Links

• Gheorghe Coserea, Rows n = 1..101, flattened
• F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.

Crossrefs

Column k=1 give A006013.

Programs

Mathematica

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 *)

PARI

N=11; x ='x + O('x^N);
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)))))

Formula

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.

P_n(0) = A006318(n-1), P_n(1) = A156017(n-1), P_n(2) = A231690(n).