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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-François 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 September 25 06:59 EDT 2023. Contains 365582 sequences. (Running on oeis4.)