A384145
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x*A(x)^3) ).
Original entry on oeis.org
1, 1, 2, 8, 44, 298, 2359, 21112, 209175, 2262121, 26431042, 331096188, 4419824468, 62565545535, 935341395343, 14716294925179, 242945752432294, 4197094127399756, 75698807290515322, 1422350601250404765, 27788515730656558613, 563512508612712699574, 11841983002490204813514
Offset: 0
-
a(n, k=1) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-2*j+k, j)/(3*n-2*j+k)*a(n-j, j)));
A384653
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 A384649.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 9, 0, 1, 4, 9, 22, 56, 0, 1, 5, 14, 40, 134, 432, 0, 1, 6, 20, 64, 240, 1012, 3935, 0, 1, 7, 27, 95, 381, 1779, 9039, 40820, 0, 1, 8, 35, 134, 565, 2780, 15596, 92246, 471633, 0, 1, 9, 44, 182, 801, 4071, 23950, 156597, 1051558, 5980210, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 5, 9, 14, 20, 27, ...
0, 9, 22, 40, 64, 95, 134, ...
0, 56, 134, 240, 381, 565, 801, ...
0, 432, 1012, 1779, 2780, 4071, 5718, ...
0, 3935, 9039, 15596, 23950, 34515, 47786, ...
-
a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-3*j+k, j)/(4*n-3*j+k)*a(n-j, j)));
A384654
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 A384650.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 10, 0, 1, 4, 9, 24, 69, 0, 1, 5, 14, 43, 162, 592, 0, 1, 6, 20, 68, 285, 1362, 6052, 0, 1, 7, 27, 100, 445, 2352, 13664, 70870, 0, 1, 8, 35, 140, 650, 3612, 23171, 157592, 928497, 0, 1, 9, 44, 189, 909, 5201, 34972, 263190, 2039543, 13404514, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 5, 9, 14, 20, 27, ...
0, 10, 24, 43, 68, 100, 140, ...
0, 69, 162, 285, 445, 650, 909, ...
0, 592, 1362, 2352, 3612, 5201, 7188, ...
0, 6052, 13664, 23171, 34972 ,49540, 67433, ...
-
a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n-4*j+k, j)/(5*n-4*j+k)*a(n-j, j)));
A384651
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 A162661.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 7, 0, 1, 4, 9, 18, 33, 0, 1, 5, 14, 34, 84, 189, 0, 1, 6, 20, 56, 159, 472, 1249, 0, 1, 7, 27, 85, 265, 882, 3057, 9237, 0, 1, 8, 35, 122, 410, 1460, 5615, 22190, 74972, 0, 1, 9, 44, 168, 603, 2256, 9166, 40053, 177149, 659042, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 5, 9, 14, 20, 27, ...
0, 7, 18, 34, 56, 85, 122, ...
0, 33, 84, 159, 265, 410, 603, ...
0, 189, 472, 882, 1460, 2256, 3330, ...
0, 1249, 3057, 5615, 9166, 14015, 20540, ...
-
a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-j+k, j)/(2*n-j+k)*a(n-j, j)));
A384681
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 A384680.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 7, 15, 0, 1, 4, 12, 36, 100, 0, 1, 5, 18, 64, 239, 805, 0, 1, 6, 25, 100, 426, 1900, 7442, 0, 1, 7, 33, 145, 671, 3357, 17319, 76750, 0, 1, 8, 42, 200, 985, 5260, 30228, 176214, 866818, 0, 1, 9, 52, 266, 1380, 7706, 46880, 303687, 1965938, 10586499, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 3, 7, 12, 18, 25, 33, ...
0, 15, 36, 64, 100, 145, 200, ...
0, 100, 239, 426, 671, 985, 1380, ...
0, 805, 1900, 3357, 5260, 7706, 10806, ...
0, 7442, 17319, 30228, 46880, 68115, 94918, ...
-
a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-j+k, j)/(3*n-j+k)*a(n-j, j)));
Showing 1-5 of 5 results.