A384581
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 A143501.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 4, 0, 1, 4, 6, 10, 16, 0, 1, 5, 10, 19, 41, 92, 0, 1, 6, 15, 32, 78, 224, 616, 0, 1, 7, 21, 50, 131, 411, 1464, 4729, 0, 1, 8, 28, 74, 205, 672, 2617, 11002, 40776, 0, 1, 9, 36, 105, 306, 1031, 4170, 19251, 93234, 388057, 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, 4, 10, 19, 32, 50, 74, ...
0, 16, 41, 78, 131, 205, 306, ...
0, 92, 224, 411, 672, 1031, 1518, ...
0, 616, 1464, 2617, 4170, 6245, 8997, ...
-
a(n, k) = 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)));
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, ...
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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)));
A384576
G.f. A(x) satisfies A(x) = ( 1 + x * A(x*A(x))^(1/2) )^2.
Original entry on oeis.org
1, 2, 3, 8, 27, 118, 609, 3602, 23866, 174186, 1383868, 11860702, 108889022, 1064691402, 11034753421, 120739899232, 1389891203976, 16781698952902, 211959646629376, 2793804347189762, 38347179124969391, 547046497259184494, 8096627908313404104
Offset: 0
-
a(n, k=2) = 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)));
Showing 1-4 of 4 results.