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 A055327 Triangle of rooted identity trees with n nodes and k leaves. 12
 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 6, 5, 1, 9, 13, 2, 1, 12, 28, 11, 1, 16, 53, 40, 3, 1, 20, 91, 109, 26, 1, 25, 146, 254, 116, 6, 1, 30, 223, 524, 387, 61, 1, 36, 326, 998, 1068, 329, 12, 1, 42, 461, 1774, 2587, 1289, 145, 1, 49, 634, 2995, 5678, 4133, 911, 25, 1, 56 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS Row lengths are 1,1,1,2,2,3,3,4,4,5,5,6,6,... LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1226 (first 70 rows) N. J. A. Sloane, Transforms Index entries for sequences related to rooted trees FORMULA G.f. satisfies A(x,y) = x*y + x*WEIGH(A(x,y)) - x. Shifts up under WEIGH transform. EXAMPLE Triangle begins: 1; 1; 1; 1, 1; 1, 2; 1, 4, 1; 1, 6, 5; 1, 9, 13, 2; 1, 12, 28, 11; 1, 16, 53, 40, 3; ... From Joerg Arndt, Aug 18 2014: (Start) The identity trees with n=6 nodes, as (preorder-) level sequences, together with their number of leaves, and an ASCII rendering, are: : : 1: [ 0 1 2 3 4 5 ] 1 : O--o--o--o--o--o : : 2: [ 0 1 2 3 4 3 ] 2 : O--o--o--o--o : .--o : : 3: [ 0 1 2 3 4 2 ] 2 : O--o--o--o--o : .--o : : 4: [ 0 1 2 3 4 1 ] 2 : O--o--o--o--o : .--o : : 5: [ 0 1 2 3 2 1 ] 3 : O--o--o--o : .--o : .--o : : 6: [ 0 1 2 3 1 2 ] 2 : O--o--o--o : .--o--o : This gives [1, 4, 1], row n=6 of the triangle. (End) PROG (PARI) WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i ))-1)} A(n)={my(v=[y]); for(n=2, n, v=concat([y], WeighMT(v))); apply(p->Vecrev(p/y), v)} { my(T=A(15)); for(n=1, #T, print(T[n])) } \\ Andrew Howroyd, Aug 28 2018 CROSSREFS Row sums give A004111. Columns 2 to 8: A002620(n-2), A055328, A055329, A055330, A055331, A055332, A055333. A regular version is A301342. Cf. A055334. Sequence in context: A127625 A124844 A133934 * A105260 A099510 A348593 Adjacent sequences: A055324 A055325 A055326 * A055328 A055329 A055330 KEYWORD nonn,tabf,eigen AUTHOR Christian G. Bower, May 12 2000 STATUS approved

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Last modified June 8 00:26 EDT 2023. Contains 363157 sequences. (Running on oeis4.)