A278463 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

1, 2, 2, 3, 9, 4, 4, 36, 44, 12, 5, 110, 355, 250, 48, 6, 300, 2010, 3480, 1644, 240, 7, 777, 9625, 32235, 35728, 12348, 1440, 8, 1960, 42056, 242200, 498512, 390880, 104544, 10080, 9, 4860, 173754, 1605744, 5466321, 7745220, 4581036, 986256, 80640, 10, 11880, 691620, 9807840, 51506490, 117711720, 123330680, 57537360, 10265760, 725760

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

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

A278458(x;t) = serreverse(A(-x;t))(-x).

A098558(n-1) = P_n(0), A032184(n) = P_n(1).

A052881(n) = T(n,n-1).

Example

A(x;t) = x + (2*t+2)*x^2/2! + (3*t^2+9*t+4)*x^3/3! + (4*t^3+36*t^2+44*t+12)*x^4/4! + ...
Triangle starts:
n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]
[1]  1;
[2]  2,       2;
[3]  3,       9,       4;
[4]  4,       36,      44,      12;
[5]  5,       110,     355,     250,     48;
[6]  6,       300,     2010,    3480,    1644,    240;
[7]  7,       777,     9625,    32235,   35728,   12348,   1440;
[8]  ...

Prog

(PARI)
N=11; x = 'x+O('x^N);
concat(apply(p->Vec(p), Vec(serlaplace((t-1)*log(1-x) - log(-x + exp(t*log(1-x))) - x))))

Crossrefs

Sequence in context: A181206 A274959 A143307 * A276248 A322891 A275329

Adjacent sequences: A278460 A278461 A278462 * A278464 A278465 A278466

Keyword

nonn,tabl

Author

Gheorghe Coserea, Jan 18 2017

Status

approved