A384896
G.f. A(x) satisfies A(x) = 1 + x/A(-x*A(x))^3.
Original entry on oeis.org
1, 1, 3, 0, -23, -51, 27, 920, 5469, 4836, -84822, -515991, -1733406, 2541688, 64653336, 324962160, 800371560, -3164656113, -49575569463, -260541998755, -734864189592, 1794936737274, 39518722602456, 260877913774320, 1122691536976305, 1485180173013631
Offset: 0
-
a(n, k=-1) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+2*j+k-1, j-1)*a(n-j, 3*j)/j));
A384903
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384898.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 6, 0, 1, 4, 12, 18, 10, 0, 1, 5, 18, 37, 41, -39, 0, 1, 6, 25, 64, 102, -22, -546, 0, 1, 7, 33, 100, 203, 96, -1074, -3563, 0, 1, 8, 42, 146, 355, 372, -1419, -8332, -18918, 0, 1, 9, 52, 203, 570, 876, -1338, -13974, -48606, -68472, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, ...
0, 3, 7, 12, 18, 25, ...
0, 6, 18, 37, 64, 100, ...
0, 10, 41, 102, 203, 355, ...
0, -39, -22, 96, 372, 876, ...
-
b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-3*n+4*j+k-1,j-1)*b(n-j,3*j)/j));
a(n, k) = b(n, -k);
A384944
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384941.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 4, 0, 1, 3, 9, -2, 0, 1, 4, 15, 4, -64, 0, 1, 5, 22, 19, -116, -95, 0, 1, 6, 30, 44, -144, -334, 780, 0, 1, 7, 39, 80, -135, -675, 862, 5230, 0, 1, 8, 49, 128, -75, -1060, 70, 11516, 19228, 0, 1, 9, 60, 189, 51, -1414, -1684, 16953, 59632, -90488, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 4, 9, 15, 22, 30, 39, ...
0, -2, 4, 19, 44, 80, 128, ...
0, -64, -116, -144, -135, -75, 51, ...
0, -95, -334, -675, -1060, -1414, -1644, ...
0, 780, 862, 70, -1684, -4380, -7869, ...
-
b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+2*j+k-1, j-1)*b(n-j, 4*j)/j));
a(n, k) = b(n, -k);
A384945
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384942.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 11, -5, 0, 1, 4, 18, 0, -135, 0, 1, 5, 26, 16, -255, -110, 0, 1, 6, 35, 44, -345, -540, 3661, 0, 1, 7, 45, 85, -389, -1230, 5777, 16440, 0, 1, 8, 56, 140, -370, -2100, 5918, 40452, -1375, 0, 1, 9, 68, 210, -270, -3049, 3784, 67356, 86065, -827075, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 5, 11, 18, 26, 35, 45, ...
0, -5, 0, 16, 44, 85, 140, ...
0, -135, -255, -345, -389, -370, -270, ...
0, -110, -540, -1230, -2100, -3049, -3954, ...
0, 3661, 5777, 5918, 3784, -770, -7708, ...
-
b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+2*j+k-1, j-1)*b(n-j, 5*j)/j));
a(n, k) = b(n, -k);
A384946
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A384943.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 6, 0, 1, 3, 13, -9, 0, 1, 4, 21, -6, -244, 0, 1, 5, 30, 10, -470, -39, 0, 1, 6, 40, 40, -660, -674, 11262, 0, 1, 7, 51, 85, -795, -1824, 19599, 36971, 0, 1, 8, 63, 146, -855, -3384, 24171, 100390, -268890, 0, 1, 9, 76, 224, -819, -5224, 24318, 180627, -268456, -3724293, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 6, 13, 21, 30, 40, 51, ...
0, -9, -6, 10, 40, 85, 146, ...
0, -244, -470, -660, -795, -855, -819, ...
0, -39, -674, -1824, -3384, -5224, -7188, ...
0, 11262, 19599, 24171, 24318, 19590, 9778, ...
-
b(n, k) = if(n*k==0, 0^n, (-1)^n*k*sum(j=1, n, binomial(-n+2*j+k-1, j-1)*b(n-j, 6*j)/j));
a(n, k) = b(n, -k);
Showing 1-5 of 5 results.