This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A286800 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 4
 1, 1, 2, 7, 6, 63, 74, 10, 729, 974, 254, 8, 10113, 15084, 5376, 406, 161935, 264724, 117424, 14954, 320, 2923135, 5163276, 2697804, 481222, 23670, 112, 58547761, 110483028, 65662932, 14892090, 1186362, 21936, 1286468225, 2570021310, 1695874928, 461501018, 51034896, 1866986, 11264, 30747331223, 64547199082, 46461697760, 14603254902, 2055851560, 116329886, 1905888, 2560 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Row n>0 contains floor(2*(n+1)/3) terms. LINKS Gheorghe Coserea, Rows n=1..123, flattened Luca G. Molinari, Nicola Manini, Enumeration of many-body skeleton diagrams, arXiv:cond-mat/0512342 [cond-mat.str-el], 2006. FORMULA y(x;t) = Sum_{n>0} P_n(t)*x^n satisfies x*deriv(y,x) = (1-y) * (2*t*x^2*(1-y)^2 + x*(1-y) - y) / (t*x^2*(1-y)^2 - t*x*y*(1-y) - 2*y), with y(0;t) = 0, where P_n(t) = Sum_{k=0..floor((2*n-1)/3)} T(n,k)*t^k for n>0. A049464(n) = T(n,0), P_n(-1) = (-1)^(n-1), A287029(n) = P_n(1). EXAMPLE A(x;t) = x + (1 + 2*t)*x^2 + (7 + 6*t)*x^3 + (63 + 74*t + 10*t^2)*x^4 + ... Triangle starts: n\k  [0]       [1]        [2]       [3]       [4]      [5] [1]  1; [2]  1,        2; [3]  7,        6; [4]  63,       74,        10; [5]  729,      974,       254,      8; [6]  10113,    15084,     5376,     406; [7]  161935,   264724,    117424,   14954,    320; [8]  2923135,  5163276,   2697804,  481222,   23670,   112; [9]  58547761, 110483028, 65662932, 14892090, 1186362, 21936; [10] ... MATHEMATICA max = 12; y0[0, _] = y1[0, _] = 0; y0[x_, t_] = x; y1[x_, t_] = 0; For[n = 1, n <= max, n++, y1[x_, t_] = Normal[(1/(-1 + y0[x, t]))*x*(-1 - y0[x, t]^2 - 2*y0[x, t]*(-1 + D[y0[x, t], x]) + t*x*(-1 + y0[x, t])*(2*(-1 + y0[x, t])^2 + (x*(-1 + y0[x, t]) + y0[x, t])*D[y0[x, t], x])) + O[x]^n]; y0[x_, t_] = y1[x, t]]; row[n_] := CoefficientList[SeriesCoefficient[y0[x, t], {x, 0, n}], t]; Flatten[Table[row[n], {n, 0, max-1}]] (* Jean-François Alcover, May 24 2017, adapted from PARI *) PROG (PARI) A286795_ser(N, t='t) = {   my(x='x+O('x^N), y0=1, y1=0, n=1);   while(n++,     y1 = (1 + x*(1 + 2*t + x*t^2)*y0^2 + t*(1-t)*x^2*y0^3 + 2*x^2*y0*y0');     y1 = y1 / (1+2*x*t); if (y1 == y0, break()); y0 = y1; ); y0; }; A286798_ser(N, t='t) = {   my(v = A286795_ser(N, t)); subst(v, 'x, serreverse(x/(1-x*t*v))); }; A286800_ser(N, t='t) = {   my(v = A286798_ser(N, t)); 1-1/subst(v, 'x, serreverse(x*v^2)); }; concat(apply(p->Vecrev(p), Vec(A286800_ser(12)))) \\ test: y=A286800_ser(50); x*y' == (1-y) * (2*t*x^2*(1-y)^2 + x*(1-y) - y) / (t*x^2*(1-y)^2 - t*x*y*(1-y) - 2*y) CROSSREFS Cf. A286781, A286782, A286783, A286784, A286785. Sequence in context: A082187 A211368 A225101 * A323722 A021365 A179378 Adjacent sequences:  A286797 A286798 A286799 * A286801 A286802 A286803 KEYWORD nonn,tabf AUTHOR Gheorghe Coserea, May 22 2017 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 15 20:50 EDT 2019. Contains 324145 sequences. (Running on oeis4.)