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A278458 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 2
1, 2, 2, 9, 15, 8, 64, 156, 144, 52, 625, 2050, 2675, 1730, 472, 7776, 32430, 55000, 50310, 25108, 5504, 117649, 599319, 1258775, 1484245, 1052184, 428036, 78416, 2097152, 12669496, 31902416, 46103680, 42064736, 24421096, 8389552, 1320064, 43046721, 301574340, 888996066, 1524644856, 1698413409, 1269814980, 625219644, 185935104, 25637824 (list; table; graph; refs; listen; history; text; internal format)
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;t) = Sum {n>=1} P_n(t)*x^n/n! satisfies x = log(y + exp(t*log(1+y))) - (t-1)*log(1+y) - y.

A006351(n) = P_n(0), A005172(n) = P_n(1), A231691(n) = P_n(2).

EXAMPLE

A(x;t) = x + (2*t+2)*x^2/2! + (9*t^2+15*t+8)*x^3/3! + (64*t^3+156*t^2+144*t+52)*x^4/4! + ...

Triangle starts:

n\k  [1]      [2]      [3]      [4]      [5]      [6]      [7]

[1]  1;

[2]  2,       2;

[3]  9,       15,      8;

[4]  64,      156,     144,     52;

[5]  625,     2050,    2675,    1730,    472;

[6]  7776,    32430,   55000,   50310,   25108,   5504;

[7]  117649,  599319,  1258775, 1484245, 1052184, 428036,  78416;

[8]  ...

MATHEMATICA

m = 10;

(Reverse[CoefficientList[#, t]]& /@ CoefficientList[InverseSeries[Log[x + Exp[t Log[1+x]]] - (t-1) Log[1+x] - x + O[x]^m], x]) Range[0, m-1]! // Rest // Flatten (* Jean-Fran├žois Alcover, Sep 28 2019 *)

PROG

(PARI)

N=10; x = 'x + O('x^N); t='t;

concat(apply(p->Vec(p), Vec(serlaplace(serreverse(log(x + exp(t*log(1+x))) - (t-1)*log(1+x) - x)))))

CROSSREFS

Column k=1 give A000169

Sequence in context: A298663 A325936 A185755 * A309705 A290604 A039796

Adjacent sequences:  A278455 A278456 A278457 * A278459 A278460 A278461

KEYWORD

nonn,tabl

AUTHOR

Gheorghe Coserea, Jan 15 2017

STATUS

approved

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Last modified May 11 18:21 EDT 2021. Contains 343808 sequences. (Running on oeis4.)