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A318110 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 2
0, 1, 1, 3, 3, 1, 26, 26, 11, 2, 367, 367, 167, 42, 5, 7142, 7142, 3352, 944, 163, 14, 176766, 176766, 84308, 25006, 4965, 638, 42, 5304356, 5304356, 2554329, 779246, 165474, 24924, 2510, 132, 186954535, 186954535, 90600599, 28120586, 6200455, 1010814, 121086, 9908, 429, 7566084686, 7566084686, 3683084984, 1156456088, 261067596, 44535120, 5829880, 574128, 39203, 1430 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Gheorghe Coserea, Rows n=0..100, flattened

Noam Zeilberger, Counting isomorphism classes of beta-normal linear lambda terms, arXiv:1509.07596 [cs.LO], 2015.

FORMULA

A(x,t) = Sum_{n>=0} P_n(t)*x^n, where P_n(t) = Sum_{k=0..n} T(n,k)*t^k, satisfies:

A = x*t/(1-A) + deriv(A,t), with A(0,t) = 0, deriv(A,x)(0,t) = 1+t (deriv(A,v) represents the derivative of A with respect to variable v).

EXAMPLE

A(x,t) = (1+t)*x + (3+3*t+t^2)*x^2 + (26+26*t+11*t^2+2*t^3)*x^3 + ...

Triangle starts:

n\k [0]       [1]       [2]      [3]      [4]     [5]     [6]    [7]  [8]

[0] 0;

[1] 1,        1;

[2] 3,        3,        1;

[3] 26,       26,       11,      2;

[4] 367,      367,      167,     42,      5;

[5] 7142,     7142,     3352,    944,     163,    14;

[6] 176766,   176766,   84308,   25006,   4965,   638,    42;

[7] 5304356,  5304356,  2554329, 779246,  165474, 24924,  2510,  132;

[8] 186954535,186954535,90600599,28120586,6200455,1010814,121086,9908,429;

[9] ...

MATHEMATICA

rows = 10; Clear[A]; A[x_, t_] = (1+t)x;

Do[A[x_, t_] = Series[x t/(1-A[x, t]) + D[A[x, t], t], {x, 0, n}, {t, 0, n}] // Normal, {n, 2 rows}];

CoefficientList[#, t]& /@ CoefficientList[A[x, t], x] /. {} -> {0} // Take[#, rows]& // Flatten (* Jean-Fran├žois Alcover, Oct 23 2018 *)

PROG

(PARI)

seq(N) = {

  my(x='x+O('x^N), t='t, F0=(1+t)*x, F1=0, n=1);

  while(n++,

    F1 = F0^2; F1 = F1 - deriv(F1, 't)/2 + deriv(F0, 't) + x*t;

    if (F1 == F0, break()); F0 = F1);

  concat([[0]], apply(Vecrev, Vec(F0)));

};

concat(seq(10))

\\ test: y=Ser(apply(p->Polrev(p, 't), seq(101)), 'x); y == x*'t/(1-y) + deriv(y, 't)

CROSSREFS

Column 0 gives A262301.

Main diagonal gives A000108(n-1) for n>0.

Second diagonal gives A032443(n-1) for n>0.

Sequence in context: A228859 A259876 A276402 * A117262 A065431 A271082

Adjacent sequences:  A318107 A318108 A318109 * A318111 A318112 A318113

KEYWORD

nonn,tabl

AUTHOR

Gheorghe Coserea, Sep 05 2018

STATUS

approved

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Last modified October 26 06:08 EDT 2021. Contains 348257 sequences. (Running on oeis4.)