A208896 Rectangular table where the g.f. of row n satisfies: R(n,x) = 1 + x*R(n,x)^n * [d/dx x/R(n,x)] for n>=0, as read by antidiagonals.

1, 1, 1, 1, 1, -2, 1, 1, -1, 9, 1, 1, 0, 3, -56, 1, 1, 1, 0, -13, 425, 1, 1, 2, 0, 0, 71, -3726, 1, 1, 3, 3, -1, 0, -461, 36652, 1, 1, 4, 9, 0, 1, 0, 3447, -397440, 1, 1, 5, 18, 19, -12, 0, 0, -29093, 4695489, 1, 1, 6, 30, 72, 0, -14, 0, 0, 273343, -59941550

Offset

0,6

Comments

The following property accounts for the zeros along the main diagonal.

The row g.f.s satisfy: [x^k] R(n,x)^(k-n+1) = [x^k] R(n,x)^(k-n) for k>=2

and thus when k=n we have [x^n] R(n,x) = [x^n] R(n,x)^0 = 0 for n>=2.

Links

Paul D. Hanna, Rows n = 0..46, flattened.

Example

Coefficients in the n-th row g.f., R(n,x), of this table begins:
n=0: [1, 1,-2,  9, -56,  425, -3726,  36652, -397440, 4695489, ...];
n=1: [1, 1,-1,  3, -13,   71,  -461,   3447,  -29093,  273343, ...];
n=2: [1, 1, 0,  0,   0,    0,     0,      0,       0,       0, ...];
n=3: [1, 1, 1,  0,  -1,    1,     0,      0,      -5,      27, ...];
n=4: [1, 1, 2,  3,   0,  -12,   -14,     43,      96,     -50, ...];
n=5: [1, 1, 3,  9,  19,    0,  -195,   -732,    -453,    6495, ...];
n=6: [1, 1, 4, 18,  72,  201,     0,  -4200,  -27984,  -91044, ...];
n=7: [1, 1, 5, 30, 175,  880,  3106,      0, -114485,-1124735, ...];
n=8: [1, 1, 6, 45, 344, 2451, 14946,  64522,       0,-3805692, ...];
n=9: [1, 1, 7, 63, 595, 5453, 45927, 331177, 1704795,       0, ...];
n=10:[1, 1, 8, 84, 944,10550,112336,1094604, 9157984,55095601, 0,...]; ...
in which the main diagonal is zeros for n>=2.
Initial row g.f.s are illustrated by the following.
R(0,x) = 1 + x*[d/dx x/R(0,x)] begins:
R(0,x) = 1 + x - 2*x^2 + 9*x^3 - 56*x^4 + 425*x^5 - 3726*x^6 +...
which satisfies: [x^k] R(0,x)^(k+1) = [x^k] R(0,x)^k for k>=2.
...
R(1,x) = 1 + x*R(1,x)*[d/dx x/R(1,x)] begins:
R(1,x) = 1 + x - x^2 + 3*x^3 - 13*x^4 + 71*x^5 - 461*x^6 + 3447*x^7 +...
which satisfies: [x^k] R(1,x)^k = [x^k] R(1,x)^(k-1) for k>=2.
...
R(2,x) = 1 + x*R(2,x)^2*[d/dx x/R(2,x)] is satisfied by:
R(2,x) = 1 + x,
which satisfies: [x^k] R(2,x)^(k-1) = [x^k] R(2,x)^(k-2) = 0 for k>=2.
...
R(3,x) = 1 + x*R(3,x)^3*[d/dx x/R(3,x)] begins:
R(3,x) = 1 + x + x^2 - x^4 + x^5 - 5*x^8 + 27*x^9 - 147*x^10 + 996*x^11 +...
which satisfies: [x^k] R(3,x)^(k-2) = [x^k] R(3,x)^(k-3) for k>=2.
...

Prog

(PARI) {T(n, k)=local(ROWn=1+x+x*O(x^k)); for(i=0, k, ROWn=1+x*ROWn^n*deriv(x/ROWn)); polcoeff(ROWn, k)}
for(n=0, 12, for(k=0, 12, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A158883 (row 0), A158882 (row 1), A208897, A208898.

Sequence in context: A264081 A061538 A123602 * A288972 A065521 A225700

Adjacent sequences: A208893 A208894 A208895 * A208897 A208898 A208899

Keyword

sign,tabl

Author

Paul D. Hanna, Mar 03 2012

Status

approved