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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 January 16 19:49 EST 2019. Contains 319206 sequences. (Running on oeis4.)