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 A001383 Number of n-node rooted trees of height at most 3. (Formerly M1107 N0422) 19
 1, 1, 1, 2, 4, 8, 15, 29, 53, 98, 177, 319, 565, 1001, 1749, 3047, 5264, 9054, 15467, 26320, 44532, 75054, 125904, 210413, 350215, 580901, 960035, 1581534, 2596913, 4251486, 6939635, 11296231, 18337815, 29692431, 47956995, 77271074, 124212966 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS a(n+1) is also the number of n-vertex graphs that do not contain a P_4, C_4, or K_4 as induced subgraph (K_4-free trivially perfect graphs, cf. A123467). - Falk Hüffner, Jan 10 2016 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS N. J. A. Sloane, Table of n, a(n) for n=0..200 INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 62 J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478. FORMULA G.f.: S[ 3 ] := x*Product (1 - x^k)^(-p(k-1)), where p(k) = number of partitions of k. a(n+1) is the Euler transform of p(n-1), where p() = A000041 is the partition function. - Franklin T. Adams-Watters, Mar 01 2006 G.f.: 1 + x*exp( Sum_{n>=1} x^n/n * Product_{k>=1} 1/(1 - x^(n*k)) ). - Paul D. Hanna, Nov 01 2012 MAPLE s[ 2 ] := x/product('1-x^i', 'i'=1..30); # G.f. for trees of ht <=2, A000041 for k from 3 to 12 do # gets g.f. for trees of ht <= 3, 4, 5, ... s[ k ] := series(x/product('(1-x^i)^coeff(s[ k-1 ], x, i)', 'i'=1..30), x, 31); od: # For Maple program see link in A000235. with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: A000041:= etr(n-> 1): a:= n->`if`(n=0, 1, etr(k-> A000041(k-1))(n-1)): seq(a(n), n=0..40);  # Alois P. Heinz, Sep 08 2008 MATHEMATICA m = 36; CoefficientList[ Series[x*Product[(1 - x^k)^(-PartitionsP[k - 1]), {k, 1, m}], {x, 0, m}], x] // Rest // Prepend[#, 1] & (* Jean-François Alcover, Jul 05 2011, after g.f. *) PROG (PARI) {a(n)=polcoeff(1+x*exp(sum(m=1, n, x^m/m/prod(k=1, n\m+1, 1-x^(m*k)+x*O(x^n)))), n)} \\ Paul D. Hanna, Nov 01 2012 CROSSREFS Cf. A000041, A001383-A001385, A034823-A034826. Sequence in context: A088532 A271364 A036621 * A217733 A208976 A278554 Adjacent sequences:  A001380 A001381 A001382 * A001384 A001385 A001386 KEYWORD nonn,easy,nice AUTHOR STATUS approved

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Last modified January 25 15:17 EST 2022. Contains 350572 sequences. (Running on oeis4.)