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A281261 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 1
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified May 11 05:25 EDT 2021. Contains 343784 sequences. (Running on oeis4.)