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 A227774 Triangular array read by rows: T(n,k) is the number of rooted identity trees with n nodes having exactly k subtrees from the root. 12
 1, 1, 1, 1, 1, 2, 1, 3, 3, 6, 5, 1, 12, 11, 2, 25, 22, 5, 52, 49, 12, 113, 104, 28, 2, 247, 232, 65, 4, 548, 513, 152, 13, 1226, 1159, 351, 34, 2770, 2619, 818, 91, 1, 6299, 5989, 1907, 225, 6, 14426, 13734, 4460, 571, 18, 33209, 31729, 10453, 1403, 57, 76851 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Row sums = A004111. LINKS Alois P. Heinz, Rows n = 1..400, flattened FORMULA G.f.: x * Product_{n>=1} (1 + y * x^n)^A004111(n). From Alois P. Heinz, Aug 25 2017: (Start) T(n,k) = Sum_{h=0..n-k} A291529(n-1,h,k). Sum_{k>=1} k * T(n,k) = A291532(n-1). (End) EXAMPLE Triangular array T(n,k) begins: n\k:   0    1    2   3   4  ... ---+--------------------------- 01 :   1; 02 :   .    1; 03 :   .    1; 04 :   .    1,   1; 05 :   .    2,   1; 06 :   .    3,   3; 07 :   .    6,   5,  1; 08 :   .   12,  11,  2; 09 :   .   25,  22,  5; 10 :   .   52,  49, 12; 11 :   .  113, 104, 28,  2; MAPLE b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,       add(binomial(b((i-1)\$2), j)*b(n-i*j, i-1), j=0..n/i)))     end: g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, expand(       add(x^j*binomial(b((i-1)\$2), j)*g(n-i*j, i-1), j=0..n/i))))     end: T:= n-> `if`(n=1, 1,     (p-> seq(coeff(p, x, k), k=1..degree(p)))(g((n-1)\$2))): seq(T(n), n=1..25);  # Alois P. Heinz, Jul 30 2013 MATHEMATICA nn=20; f[x_]:=Sum[a[n]x^n, {n, 0, nn}]; sol=SolveAlways[0==Series[f[x]-x Product[(1+x^i)^a[i], {i, 1, nn}], {x, 0, nn}], x]; A004111=Drop[ Flatten[Table[a[n], {n, 0, nn}]/.sol], 1]; Map[Select[#, #>0&]&, Drop[CoefficientList[Series[x Product[(1 + y x^i)^A004111[[i]], {i, 1, nn}], {x, 0, nn}], {x, y}], 1]]//Grid PROG (Python) from sympy import binomial, Poly, Symbol from sympy.core.cache import cacheit x=Symbol('x') @cacheit def b(n, i):return 1 if n==0 else 0 if i<1 else sum([binomial(b(i - 1, i - 1), j)*b(n - i*j, i - 1) for j in range(n//i + 1)]) @cacheit def g(n, i):return 1 if n==0 else 0 if i<1 else sum([x**j*binomial(b(i - 1, i - 1), j)*g(n - i*j, i - 1) for j in range(n//i + 1)]) def T(n): return [1] if n==1 else Poly(g(n - 1, n - 1)).all_coeffs()[::-1][1:] for n in range(1, 26): print (T(n)) # Indranil Ghosh, Aug 28 2017 CROSSREFS Columns k=1-10 give: A004111(n-1), A227806, A227807, A227808, A227809, A227810, A227811, A227812, A227813, A227814. Cf. A291529, A291532. Sequence in context: A213934 A124774 A056610 * A214920 A096373 A216961 Adjacent sequences:  A227771 A227772 A227773 * A227775 A227776 A227777 KEYWORD nonn,tabf,look AUTHOR Geoffrey Critzer, Jul 30 2013 STATUS approved

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Last modified October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)