A153298
G.f.: A(x) = F(x*G(x)^3)^2 = F(G(x)-1)^2 where F(x) = G(x/F(x)) = 1 + x*F(x)^2 is the g.f. of A000108 (Catalan) and G(x) = F(x*G(x)) = 1 + x*G(x)^3 is the g.f. of A001764.
Original entry on oeis.org
1, 2, 11, 68, 443, 2974, 20361, 141356, 991738, 7015814, 49967892, 357896120, 2575844046, 18616823352, 135051785186, 982949932092, 7175591019313, 52524480778590, 385429134781530, 2834791998208500, 20893844524709649
Offset: 0
G.f.: A(x) = F(x*G(x)^3)^2 = 1 + 2*x + 11*x^2 + 68*x^3 + 443*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
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{a(n)=if(n==0,1,sum(k=0,n,binomial(2*k+2,k)*2/(2*k+2)*binomial(3*(n-k)+3*k,n-k)*3*k/(3*(n-k)+3*k)))}
A153390
G.f.: A(x) = F(x*G(x))^2 where F(x) = G(x*F(x)) = 1 + x*F(x)^3 is the g.f. of A001764 and G(x) = F(x/G(x)) = 1 + x*G(x)^2 is the g.f. of A000108 (Catalan).
Original entry on oeis.org
1, 2, 9, 48, 278, 1696, 10736, 69886, 465019, 3149476, 21643433, 150554144, 1058101315, 7502183626, 53599160532, 385494328218, 2788827078507, 20280590381098, 148167425970522, 1087007419753186, 8004683588800899
Offset: 0
G.f.: A(x) = F(x*G(x))^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 278*x^4 +... where
F(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
F(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
F(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 + 7752*x^6 +...
G(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
G(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
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{a(n)=if(n==0,1,sum(k=0,n,binomial(3*k+2,k)*2/(3*k+2)*binomial(2*(n-k)+k,n-k)*k/(2*(n-k)+k)))}
A381861
G.f. A(x) satisfies A(x) = (1 + x*A(x))^4 * C(x), where C(x) is the g.f. of A000108.
Original entry on oeis.org
1, 5, 32, 231, 1797, 14715, 125064, 1093194, 9766783, 88793815, 818832674, 7640868924, 72014955566, 684551660324, 6555290711728, 63179148757584, 612376024087047, 5965515657187437, 58375460484257734, 573545171374958628, 5655759227878768987, 55957005428512022905
Offset: 0
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a(n) = sum(k=0, n, binomial(n+k+1, k)*binomial(4*n-4*k+4, n-k)/(n+k+1));
Showing 1-3 of 3 results.