A120013 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.

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, -143, 1, 7, 36, 120, 192, -87, -664, -572, 1, 8, 49, 210, 530, 360, -1575, -3514, -2210, 1, 9, 64, 336, 1164, 1955, -1568, -12240, -16916, -8398, 1, 10, 81, 504, 2226, 5892, 3785, -26804, -77730, -77388, -31654

Offset

1,5

Links

Table of n, a(n) for n=1..66.

Formula

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.

Example

Square table begins:
1, 1, 1, 0, -6, -33, -143, -572, -2210, -8398, ...
1, 2, 4, 6, -4, -100, -664, -3514, -16916, -77388, ...
1, 3, 9, 24, 42, -87, -1575, -12240, -77730, -449994, ...
1, 4, 16, 60, 192, 360, -1568, -26804, -240800, -1804456, ...
1, 5, 25, 120, 530, 1955, 3785, -28900, -508610, -5227110, ...
1, 6, 36, 210, 1164, 5892, 24552, 48258, -577380, -10814388, ...
1, 7, 49, 336, 2226, 13965, 79681, 370216, 733054, -12716578, ...
1, 8, 64, 504, 3872, 28688, 200960, 1276760, 6548320, 13015536, ...
1, 9, 81, 720, 6282, 53415, 437697, 3387636, 23729310, 133234434, ...
1, 10, 100, 990, 9660, 92460, 862120, 7743550, 65644780, 502780580,...
Successive self-compositions of F(x), the g.f. of A120009, start:
F(x) = x + x^2 + x^3 - 6x^5 - 33x^6 - 143x^7 - 572x^8 - 2210x^9 +...
F(F(x)) = x + 2x^2 + 4x^3 + 6x^4 - 4x^5 - 100x^6 - 664x^7 +...
F(F(F(x))) = x + 3x^2 + 9x^3 + 24x^4 + 42x^5 - 87x^6 - 1575x^7 +...
F(F(F(F(x)))) = x + 4x^2 + 16x^3 + 60x^4 + 192x^5 + 360x^6 +...
F(F(F(F(F(x))))) = x + 5x^2 + 25x^3 + 120x^4 + 530x^5 +1955x^6 +...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 36x^3 +210x^4 +1164x^5 + 5892x^6 +...

Prog

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

Crossrefs

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

Sequence in context: A242884 A197219 A300508 * A344180 A151847 A349593

Adjacent sequences: A120010 A120011 A120012 * A120014 A120015 A120016

Keyword

sign,tabl

Author

Paul D. Hanna, Jun 10 2006

Status

approved