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)));
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)));
A384575
G.f. A(x) satisfies A(x) = 1 + x * A(x*A(x)^5).
Original entry on oeis.org
1, 1, 1, 6, 31, 236, 2166, 22722, 269889, 3567412, 51765431, 816476196, 13892821878, 253442895075, 4930644856063, 101830536332051, 2223767436058566, 51172807259226084, 1237092039069090235, 31332521053777095784, 829389782837272248191, 22894754438382163120136
Offset: 0
-
a(n, k=1) = 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)));
A384579
G.f. A(x) satisfies A(x) = ( 1 + x * A(x*A(x))^(1/5) )^5.
Original entry on oeis.org
1, 5, 15, 60, 320, 2151, 17985, 176610, 1985755, 25116455, 351852746, 5393800690, 89651625560, 1603780929525, 30688985052200, 624900202917151, 13480450499067220, 306905745410816990, 7349620218635161140, 184589745041317074895, 4849519725067777296866
Offset: 0
-
a(n, k=5) = 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)));
Showing 1-5 of 5 results.