OFFSET
0,13
COMMENTS
From Vaclav Kotesovec, Aug 26 2014: (Start)
Column k > 0 is asymptotic to c(k) * d(k)^n / n^(3/2), where constants c(k) and d(k) are dependent only on k. Conjecture: d(k) ~ k * exp(1). Numerically:
d(1) = 2.9557652856519949747148175... (A051491)
d(2) = 5.6465426162329497128927135... (A245870)
d(3) = 8.3560268792959953682760695...
d(4) = 11.0699628777593263124193026...
d(5) = 13.7856511100846851989303249...
d(6) = 16.5022088446930015657112211...
d(7) = 19.2192613290638657575973462...
d(8) = 21.9366222112987115910888213...
d(9) = 24.6541883249893084812976812...
d(10) = 27.3718979186642404090999595...
d(100) = 272.0126359583480733207362718...
d(101) = 274.7309127032967881125015217...
d(200) = 543.8405620978790523837823296...
d(201) = 546.5588426492458787468860222...
d(101)-d(100) = 2.718276744...
d(201)-d(200) = 2.718280551...
(End)
LINKS
Alois P. Heinz, Antidiagonals n = 0..140, flattened
L. Foissy, Algebraic structures on typed decorated rooted trees, arXiv:1811.07572 (2018)
FORMULA
G.f. for column k: x*Product_{n>=1} 1/(1 - x^n)^(k*A(n,k)). - Geoffrey Critzer, Nov 13 2014
EXAMPLE
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, 7, ...
0, 2, 7, 15, 26, 40, 57, 77, ...
0, 4, 26, 82, 188, 360, 614, 966, ...
0, 9, 107, 495, 1499, 3570, 7284, 13342, ...
0, 20, 458, 3144, 12628, 37476, 91566, 195384, ...
0, 48, 2058, 20875, 111064, 410490, 1200705, 2984142, ...
MAPLE
with(numtheory):
A:= proc(n, k) option remember; `if`(n<2, n, (add(add(d*
A(d, k), d=divisors(j))*A(n-j, k)*k, j=1..n-1))/(n-1))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
nn = 10; t[x_] := Sum[a[n] x^n, {n, 1, nn}]; Transpose[ Table[Flatten[ sol = SolveAlways[ 0 == Series[ t[x] - x Product[1/(1 - x^i)^(n a[i]), {i, 1, nn}], {x, 0, nn}], x]; Flatten[{0, Table[a[n], {n, 1, nn}]}] /. sol], {n, 0, nn}]] // Grid (* Geoffrey Critzer, Nov 11 2014 *)
A[n_, k_] := A[n, k] = If[n<2, n, Sum[Sum[d*A[d, k], {d, Divisors[j]}] *A[n-j, k]*k, {j, 1, n-1}]/(n-1)]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 04 2014, translated from Maple *)
PROG
(PARI) \\ ColGf gives column generating function
ColGf(N, k) = {my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = k/n * sum(i=1, n, sumdiv(i, d, d*A[d]) * A[n-i+1] ) ); x*Ser(A)}
Mat(vector(8, k, concat(0, Col(ColGf(7, k-1))))) \\ Andrew Howroyd, May 12 2018
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, May 09 2014
STATUS
approved