A120013 - OEIS

%I #6 Mar 30 2012 18:36:57

%S 1,1,1,1,2,1,1,3,4,0,1,4,9,6,-6,1,5,16,24,-4,-33,1,6,25,60,42,-100,

%T -143,1,7,36,120,192,-87,-664,-572,1,8,49,210,530,360,-1575,-3514,

%U -2210,1,9,64,336,1164,1955,-1568,-12240,-16916,-8398,1,10,81,504,2226,5892,3785,-26804,-77730,-77388,-31654

%N Square table, read by antidiagonals, of coefficients of x^k in the n-th self-composition of the g.f. of A120009, so that: T(n,k) = [x^k] { (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2 } for n>=1, k>=1.

%F T(n,k) = Sum_{j=1..k} n^(j-2)*(n-j+1)*j*(2*k-j-1)!/(k-j)!/k! - Paul D. Hanna and _Max Alekseyev_. Alternate formula: T(n,k) = n^(k-1) - Sum_{j=2..k-2} n^(j-1)*j*(j-1)*(k-j-1)*(2*k-j-2)!/(k-j)!/k!. These formulas also apply to non-integer n.

%e Square table begins:

%e 1, 1, 1, 0, -6, -33, -143, -572, -2210, -8398, ...

%e 1, 2, 4, 6, -4, -100, -664, -3514, -16916, -77388, ...

%e 1, 3, 9, 24, 42, -87, -1575, -12240, -77730, -449994, ...

%e 1, 4, 16, 60, 192, 360, -1568, -26804, -240800, -1804456, ...

%e 1, 5, 25, 120, 530, 1955, 3785, -28900, -508610, -5227110, ...

%e 1, 6, 36, 210, 1164, 5892, 24552, 48258, -577380, -10814388, ...

%e 1, 7, 49, 336, 2226, 13965, 79681, 370216, 733054, -12716578, ...

%e 1, 8, 64, 504, 3872, 28688, 200960, 1276760, 6548320, 13015536, ...

%e 1, 9, 81, 720, 6282, 53415, 437697, 3387636, 23729310, 133234434, ...

%e 1, 10, 100, 990, 9660, 92460, 862120, 7743550, 65644780, 502780580,...

%e Successive self-compositions of F(x), the g.f. of A120009, start:

%e F(x) = x + x^2 + x^3 - 6x^5 - 33x^6 - 143x^7 - 572x^8 - 2210x^9 +...

%e F(F(x)) = x + 2x^2 + 4x^3 + 6x^4 - 4x^5 - 100x^6 - 664x^7 +...

%e F(F(F(x))) = x + 3x^2 + 9x^3 + 24x^4 + 42x^5 - 87x^6 - 1575x^7 +...

%e F(F(F(F(x)))) = x + 4x^2 + 16x^3 + 60x^4 + 192x^5 + 360x^6 +...

%e F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 120x^4 + 530x^5 +1955x^6 +...

%e F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 +210x^4 +1164x^5 + 5892x^6 +...

%o (PARI) {T(n,k)=sum(j=1,k,n^(j-2)*(n-j+1)*j*(2*k-j-1)!/(k-j)!/k!)}

%Y Rows: A120009, A127275, A120012; Diagonals: A120014, A120015.

%K sign,tabl

%O 1,5

%A _Paul D. Hanna_, Jun 10 2006