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)));
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)));
A384574
G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^4).
Original entry on oeis.org
1, 1, 1, 5, 23, 155, 1236, 11286, 116333, 1329433, 16630343, 225606826, 3294976854, 51496560764, 856858516809, 15112857079891, 281479726839851, 5517842789917283, 113510479973132860, 2444032094604379100, 54948814775692303024, 1287258966133883349701
Offset: 0
-
a(n, k=1) = 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)));
A384578
G.f. A(x) satisfies A(x) = ( 1 + x * A(x*A(x))^(1/4) )^4.
Original entry on oeis.org
1, 4, 10, 36, 171, 1032, 7656, 66144, 651065, 7170044, 87058242, 1152623008, 16497960553, 253521890800, 4158356425944, 72446946779420, 1335030266607501, 25927404824529616, 528984983237731754, 11306375975258492540, 252529515598101796399, 5880779189553142120704
Offset: 0
-
a(n, k=4) = 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)));
Showing 1-5 of 5 results.