A295811 G.f. A(x) satisfies: [x^(n-1)] A(x)^(n^2) = 2*n * [x^(n-2)] A(x)^(n^2) for n>=2, with A(0) = 1.

1, 1, 2, 11, 140, 2898, 80844, 2786091, 113184008, 5266198778, 275248731860, 15939117549502, 1012084698990904, 69901132180300132, 5217426460077854712, 418615099531669351443, 35942031310982080239120, 3289533291926922095871546, 319841125714352173292953668, 32937612567848507536114539402, 3582858531960091228861488651864

Offset: 0

Author

Paul D. Hanna, Feb 02 2018

Keywords

• nonn

Comments

Compare g.f. to: [x^(n-1)] G(x)^n = 2 * [x^(n-2)] G(x)^n for n>=2 holds when G(x) = 1/(1-x).

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 140*x^4 + 2898*x^5 + 80844*x^6 + 2786091*x^7 + 113184008*x^8 + 5266198778*x^9 + 275248731860*x^10 + ...
ILLUSTRATION OF THE DEFINITION.
The table of coefficients of x^k in A(x)^(n^2) begins:
n=1: [1, 1, 2, 11, 140, 2898, 80844, ...];
n=2: [1, 4, 14, 72, 741, 13724, 364546, ...];
n=3: [1, 9, 54, 327, 2826, 42660, 1017720, ...];
n=4: [1, 16, 152, 1216, 10540, 129376, 2559792, ...];
n=5: [1, 25, 350, 3775, 37750, 427480, 6820800, ...];
n=6: [1, 36, 702, 10056, 123165, 1477980, 20712546, ...];
n=7: [1, 49, 1274, 23667, 359856, 4953998, 69355972, ...]; ...
in which the main diagonal
D0 = [1, 4, 54, 1216, 37750, 1477980, 69355972, 3775816704, ...]
and the adjacent diagonal
D1 = [1, 9, 152, 3775, 123165, 4953998, 235988544, 12954335103, ...]
are related by D0[n-1] = 2*n*D1[n-2] for n>=2.
The related sequence D0[n-1]/n^2, n>=1, begins:
[1, 1, 6, 76, 1510, 41055, 1415428, 58997136, 2878741134, 160698224230, ...].
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A295766, A088715, A300873.

Programs

PARI

{a(n) = my(A=[1]); for(m=2, n+1, A=concat(A, 0); V=Vec(Ser(A)^(m^2)); A[#A] = V[#A-1]*2/m - V[#A]/m^2 ); A[n+1]}
for(n=0,20,print1(a(n),", "))

Formula

a(2^k - 1) is odd for k>=0 and a(n) is even elsewhere (conjecture).

a(n) ~ c * d^n * n! / n^3, where d = -4/(LambertW(-2*exp(-2))*(2+LambertW(-2*exp(-2)))) = 6.176554609483480358231680164050876553672889794284... and c = 2.719099850893334482... - Vaclav Kotesovec, Feb 07 2018