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 A242249 Number A(n,k) of rooted trees with n nodes and k-colored non-root nodes; square array A(n,k), n>=0, k>=0, read by antidiagonals. 16
 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 2, 0, 0, 1, 3, 7, 4, 0, 0, 1, 4, 15, 26, 9, 0, 0, 1, 5, 26, 82, 107, 20, 0, 0, 1, 6, 40, 188, 495, 458, 48, 0, 0, 1, 7, 57, 360, 1499, 3144, 2058, 115, 0, 0, 1, 8, 77, 614, 3570, 12628, 20875, 9498, 286, 0, 0, 1, 9, 100, 966, 7284, 37476, 111064, 142773, 44947, 719, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,13 COMMENTS From Vaclav Kotesovec, Aug 26 2014: (Start) Column k > 0 is asymptotic to c(k) * d(k)^n / n^(3/2), where constants c(k) and d(k) are dependent only on k. Conjecture: d(k) ~ k * exp(1). Numerically: d(1)   =   2.9557652856519949747148175... (A051491) d(2)   =   5.6465426162329497128927135... (A245870) d(3)   =   8.3560268792959953682760695... d(4)   =  11.0699628777593263124193026... d(5)   =  13.7856511100846851989303249... d(6)   =  16.5022088446930015657112211... d(7)   =  19.2192613290638657575973462... d(8)   =  21.9366222112987115910888213... d(9)   =  24.6541883249893084812976812... d(10)  =  27.3718979186642404090999595... d(100) = 272.0126359583480733207362718... d(101) = 274.7309127032967881125015217... d(200) = 543.8405620978790523837823296... d(201) = 546.5588426492458787468860222... d(101)-d(100) = 2.718276744... d(201)-d(200) = 2.718280551... (End) LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 (2018) FORMULA G.f. for column k: x*Product_{n>=1} 1/(1 - x^n)^(k*A(n,k)). - Geoffrey Critzer, Nov 13 2014 EXAMPLE Square array A(n,k) begins:   0,  0,    0,     0,      0,      0,       0,       0, ...   1,  1,    1,     1,      1,      1,       1,       1, ...   0,  1,    2,     3,      4,      5,       6,       7, ...   0,  2,    7,    15,     26,     40,      57,      77, ...   0,  4,   26,    82,    188,    360,     614,     966, ...   0,  9,  107,   495,   1499,   3570,    7284,   13342, ...   0, 20,  458,  3144,  12628,  37476,   91566,  195384, ...   0, 48, 2058, 20875, 111064, 410490, 1200705, 2984142, ... MAPLE with(numtheory): A:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*       A(d, k), d=divisors(j))*A(n-j, k)*k, j=1..n-1))/(n-1))     end: seq(seq(A(n, d-n), n=0..d), d=0..12); MATHEMATICA nn = 10; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; Transpose[ Table[Flatten[ sol = SolveAlways[ 0 == Series[ t[x] - x Product[1/(1 - x^i)^(n a[i]), {i, 1, nn}], {x, 0, nn}], x]; Flatten[{0, Table[a[n], {n, 1, nn}]}] /. sol], {n, 0, nn}]] // Grid (* Geoffrey Critzer, Nov 11 2014 *) A[n_, k_] := A[n, k] = If[n<2, n, Sum[Sum[d*A[d, k], {d, Divisors[j]}] *A[n-j, k]*k, {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 04 2014, translated from Maple *) PROG (PARI) \\ ColGf gives column generating function ColGf(N, k) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = k/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); x*Ser(A)} Mat(vector(8, k, concat(0, Col(ColGf(7, k-1))))) \\ Andrew Howroyd, May 12 2018 CROSSREFS Columns k=0-10 give: A063524, A000081, A000151, A006964, A052763, A052788, A246235, A246236, A246237, A246238, A246239. Rows n=0-3 give: A000004, A000012, A001477, A005449. Lower diagonal gives A242375. Cf. A255517, A256064. Sequence in context: A136438 A059848 A036865 * A125226 A281790 A245254 Adjacent sequences:  A242246 A242247 A242248 * A242250 A242251 A242252 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, May 09 2014 STATUS approved

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Last modified December 5 20:45 EST 2019. Contains 329779 sequences. (Running on oeis4.)