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A227774
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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.
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12
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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
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OFFSET
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1,6
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COMMENTS
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LINKS
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FORMULA
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G.f.: x * Product_{n>=1} (1 + y * x^n)^A004111(n).
T(n,k) = Sum_{h=0..n-k} A291529(n-1,h,k).
Sum_{k>=1} k * T(n,k) = A291532(n-1). (End)
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EXAMPLE
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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;
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MAPLE
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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))):
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MATHEMATICA
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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
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PROG
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(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:]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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