A281261 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

1, 2, 2, 1, 5, 2, 5, 9, 2, 1, 15, 14, 2, 7, 35, 20, 2, 1, 28, 70, 27, 2, 9, 84, 126, 35, 2, 1, 45, 210, 210, 44, 2, 11, 165, 462, 330, 54, 2, 1, 66, 495, 924, 495, 65, 2, 13, 286, 1287, 1716, 715, 77, 2, 1, 91, 1001, 3003, 3003, 1001, 90, 2, 15, 455, 3003, 6435, 5005, 1365, 104, 2, 1, 120, 1820, 8008, 12870, 8008, 1820, 119, 2

Offset

1,2

Comments

Row n>1 contains floor((n+3)/2) terms.

Links

Gheorghe Coserea, Rows n = 1..202, 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

A(x;t) = Sum{n>=1} P_n(t)*x^n = x*((1-t)*x^3 + (t^2-2*t-1)*x^2 + (2*t-1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1).

A278457(x;t) = serreverse(A(-x;t))(-x).

A151821(n) = P_n(1), A213667(n) = P_n(2).

P_n(t^2) = ((1+t)^(n+1) + (1-t)^(n+1))/2 - t^2 + 1, for n>1.

Example

A(x;t) = x + (2*t+2)*x^2 + (t^2+5*t+2)*x^3 + (5*t^2+9*t+2)*x^4 + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]      [8]
[1]  1;
[2]  2,       2;
[3]  1,       5,       2;
[4]  5,       9,       2;
[5]  1,       15,      14,      2;
[6]  7,       35,      20,      2;
[7]  1,       28,      70,      27,      2;
[8]  9,       84,      126,     35,      2;
[9]  1,       45,      210,     210,     44,      2;
[10] 11,      165,     462,     330,     54,      2;
[11] 1,       66,      495,     924,     495,     65,      2;
[12] 13,      286,     1287,    1716,    715,     77,      2;
[13] 1,       91,      1001,    3003,    3003,    1001,    90,      2;
[14] 15,      455,     3003,    6435,    5005,    1365,    104,     2;
[15] ...

Mathematica

Reverse[CoefficientList[#, t]]& /@ CoefficientList[x*((1-t)*x^3 + (t^2 - 2*t - 1)*x^2 + (2*t - 1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1) + O[x]^16, x] // Rest // Flatten (* Jean-François Alcover, Feb 18 2019 *)

Prog

(PARI)
N=16; x='x+O('x^N); concat(apply(p->Vec(p),  Vec(Ser(x*((1-t)*x^3 + (t^2-2*t-1)*x^2 + (2*t-1)*x + 1)/((t-1)*x^3 + (3-t)*x^2 - 3*x + 1)))))
(PARI)
N = 14; concat(1, concat(vector(N, n, Vec(substpol(((1+t)^(n+2) + (1-t)^(n+2))/2 - t^2 + 1, t^2, t)))))

Crossrefs

Sequence in context: A123398 A277495 A188945 * A102849 A088333 A016538

Adjacent sequences: A281258 A281259 A281260 * A281262 A281263 A281264

Keyword

nonn,tabf

Author

Gheorghe Coserea, Jan 18 2017

Status

approved