A384652
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 A384145.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 5, 8, 0, 1, 4, 9, 20, 44, 0, 1, 5, 14, 37, 108, 298, 0, 1, 6, 20, 60, 198, 716, 2359, 0, 1, 7, 27, 90, 321, 1290, 5554, 21112, 0, 1, 8, 35, 128, 485, 2064, 9821, 48838, 209175, 0, 1, 9, 44, 175, 699, 3091, 15452, 84888, 476714, 2262121, 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, 8, 20, 37, 60, 90, 128, ...
0, 44, 108, 198, 321, 485, 699, ...
0, 298, 716, 1290, 2064, 3091, 4434, ...
0, 2359, 5554, 9821, 15452, 22805, 32315, ...
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a(n, k) = 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)));
A384649
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x*A(x)^4) ).
Original entry on oeis.org
1, 1, 2, 9, 56, 432, 3935, 40820, 471633, 5980210, 82329140, 1220547845, 19359684220, 326799737576, 5844913732057, 110341722975077, 2191461358459051, 45656013573862832, 995196646595460516, 22644288881875546322, 536706817952488705651, 13225669497771610891404
Offset: 0
-
a(n, k=1) = 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)));
A384650
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x*A(x)^5) ).
Original entry on oeis.org
1, 1, 2, 10, 69, 592, 6052, 70870, 928497, 13404514, 210892157, 3584892350, 65390514877, 1272723903336, 26307949481077, 575201364472316, 13255835789428863, 320999903683710948, 8145524458876305526, 216062918679078474529, 5977572987203090333399
Offset: 0
-
a(n, k=1) = 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)));
A384680
G.f. A(x) satisfies A(x) = 1/( 1 - x*A(x)*A(x*A(x)^3) ).
Original entry on oeis.org
1, 1, 3, 15, 100, 805, 7442, 76750, 866818, 10586499, 138549918, 1929878820, 28459172110, 442421488758, 7225177328165, 123586748434192, 2208493015533530, 41138303109509415, 797178212982793708, 16041390159326400966, 334654194086236031816, 7227174934846895031544
Offset: 0
-
terms = 22; A[] = 0; Do[A[x] = 1/(1-x*A[x]*A[x*A[x]^3]) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jun 07 2025 *)
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a(n, k=1) = 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 1-4 of 4 results.