A384582
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 A384574.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 5, 0, 1, 4, 6, 12, 23, 0, 1, 5, 10, 22, 57, 155, 0, 1, 6, 15, 36, 105, 366, 1236, 0, 1, 7, 21, 55, 171, 651, 2853, 11286, 0, 1, 8, 28, 80, 260, 1032, 4951, 25584, 116333, 0, 1, 9, 36, 112, 378, 1536, 7656, 43587, 259789, 1329433, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 3, 6, 10, 15, 21, ...
0, 5, 12, 22, 36, 55, 80, ...
0, 23, 57, 105, 171, 260, 378, ...
0, 155, 366, 651, 1032, 1536, 2196, ...
0, 1236, 2853, 4951, 7656, 11125, 15552, ...
-
a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(4*n-4*j+k, j)/(4*n-4*j+k)*a(n-j, j)));
A384583
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 A384575.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 6, 0, 1, 4, 6, 14, 31, 0, 1, 5, 10, 25, 75, 236, 0, 1, 6, 15, 40, 135, 546, 2166, 0, 1, 7, 21, 60, 215, 951, 4902, 22722, 0, 1, 8, 28, 86, 320, 1476, 8338, 50620, 269889, 0, 1, 9, 36, 119, 456, 2151, 12634, 84714, 593347, 3567412, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 3, 6, 10, 15, 21, ...
0, 6, 14, 25, 40, 60, 86, ...
0, 31, 75, 135, 215, 320, 456, ...
0, 236, 546, 951, 1476, 2151, 3012, ...
0, 2166, 4902, 8338, 12634, 17985, 24627, ...
-
a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(5*n-5*j+k, j)/(5*n-5*j+k)*a(n-j, j)));
A384580
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 A143500.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 3, 0, 1, 4, 6, 8, 10, 0, 1, 5, 10, 16, 27, 46, 0, 1, 6, 15, 28, 54, 118, 244, 0, 1, 7, 21, 45, 95, 228, 609, 1481, 0, 1, 8, 28, 68, 155, 392, 1144, 3602, 10020, 0, 1, 9, 36, 98, 240, 631, 1916, 6597, 23866, 74400, 0
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 1, 3, 6, 10, 15, 21, ...
0, 3, 8, 16, 28, 45, 68, ...
0, 10, 27, 54, 95, 155, 240, ...
0, 46, 118, 228, 392, 631, 972, ...
0, 244, 609, 1144, 1916, 3015, 4560, ...
-
a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(2*n-2*j+k, j)/(2*n-2*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)));
A384577
G.f. A(x) satisfies A(x) = ( 1 + x * A(x*A(x))^(1/3) )^3.
Original entry on oeis.org
1, 3, 6, 19, 78, 411, 2617, 19251, 160254, 1482400, 15035622, 165545253, 1963006576, 24908182305, 336397711074, 4813816122917, 72704962269990, 1155070280657286, 19245587072017468, 335418172582313610, 6100293082529588802, 115532044092709366555, 2274095852526512246841
Offset: 0
-
a(n, k=3) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+k, j)/(3*n-3*j+k)*a(n-j, j)));
Showing 1-5 of 5 results.