A364339
G.f. satisfies A(x) = (1 + x) * (1 + x*A(x)^6).
Original entry on oeis.org
1, 2, 13, 150, 1978, 28603, 438273, 6992052, 114915180, 1932233883, 33081722359, 574755965137, 10107627041697, 179576579730534, 3218352405778284, 58114340679967608, 1056284029850962674, 19310039426151335622, 354818596435147647654, 6549556302551204621664, 121394125733645986376838
Offset: 0
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terms = 21; A[] = 0; Do[A[x] = (1+x)(1+x*A[x]^6) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Mar 24 2025 *)
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a(n) = sum(k=0, n, binomial(6*k+1, k)*binomial(6*k+1, n-k)/(6*k+1));
A364331
G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^5).
Original entry on oeis.org
1, 2, 15, 163, 2070, 28698, 421015, 6425644, 100977137, 1622885389, 26551709946, 440744175801, 7404449354076, 125657625548824, 2150963575012295, 37094953102567208, 643904274979347286, 11241232087809137759, 197247501440314516840, 3476787208220672891388, 61533794803235280779261
Offset: 0
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A364331 := proc(n)
add( binomial(2*n+3*k+1,k) * binomial(2*n+3*k+1,n-k)/(2*n+3*k+1),k=0..n) ;
end proc:
seq(A364331(n),n=0..70); # R. J. Mathar, Jul 25 2023
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a(n) = sum(k=0, n, binomial(2*n+3*k+1, k)*binomial(2*n+3*k+1, n-k)/(2*n+3*k+1));
A379280
G.f. A(x) satisfies A(x) = ( (1 + x*A(x)^3) * (1 + x*A(x)) )^2.
Original entry on oeis.org
1, 4, 38, 500, 7601, 125520, 2187736, 39608616, 737651032, 14040612502, 271931510448, 5341639974490, 106167131932708, 2131125360950758, 43142742495766252, 879810600033569754, 18057207334571432048, 372701480245014988624, 7731178967720860156743
Offset: 0
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a(n) = sum(k=0, n, binomial(2*n+4*k+2, k)*binomial(2*n+4*k+2, n-k)/(n+2*k+1));
A364340
G.f. satisfies A(x) = (1 + x*A(x)) * (1 + x*A(x)^6).
Original entry on oeis.org
1, 2, 15, 179, 2502, 38262, 619991, 10459410, 181771289, 3231782239, 58505593456, 1074766446526, 19984671314164, 375414901633692, 7113886504446443, 135820770971898805, 2610186429457347486, 50452256583633573513, 980187901557594671335, 19130197594133100828170, 374894511736219913097375
Offset: 0
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a(n) = sum(k=0, n, binomial(n+5*k+1, k)*binomial(n+5*k+1, n-k)/(n+5*k+1));
Showing 1-4 of 4 results.