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

1,2

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.

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).

Example

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]  ...

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 *)

Prog

(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)))))

Crossrefs

Column k=1 give A006013.

Sequence in context: A000024 A005223 A118680 * A094116 A079476 A005298

Adjacent sequences: A278454 A278455 A278456 * A278458 A278459 A278460

Keyword

nonn,tabl

Author

Gheorghe Coserea, Jan 15 2017

Status

approved