A137571
Main diagonal of square array A137570.
Original entry on oeis.org
1, 2, 10, 60, 397, 2802, 20710, 158428, 1244413, 9980220, 81394123, 672998498, 5628741195, 47535483498, 404790717079, 3471892750622, 29966295451511, 260080708564964, 2268416956569463, 19872441881999354, 174783803353387498
Offset: 0
G.f.: A(x) = 1 + 2*x + 10*x^2 + 60*x^3 + 397*x^4 + 2802*x^5 +...;
A(x) = 1/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].
-
{a(n)=local(m=n+1,C,F,A); C=Ser(vector(m,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))); A=1/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}
A137572
The first upper diagonal of square array A137570; equals the convolution of the main diagonal A137571 with A002293.
Original entry on oeis.org
1, 3, 16, 100, 681, 4908, 36842, 285158, 2260257, 18257902, 149769225, 1244277499, 10448404901, 88538107802, 756153001241, 6501989278168, 56244305146039, 489111092027854, 4273491476147117, 37496699100314116, 330261353255659842
Offset: 0
G.f.: A(x) = 1 + 3*x + 16*x^2 + 100*x^3 + 681*x^4 + 4908*x^5 +...;
A(x) = F(x)/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].
-
{a(n)=local(m=n+1,C,F,A); C=Ser(vector(m,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))); A=F/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}
A137573
The first lower diagonal in square array A137570; equals the convolution of the main diagonal A137571 with the Catalan numbers (A000108) and with the square of A002293.
Original entry on oeis.org
1, 5, 29, 186, 1281, 9294, 70109, 544833, 4333381, 35108351, 288738813, 2404256945, 20228988678, 171716799066, 1468804301441, 12647321103329, 109538312419238, 953622158606749, 8340394595266367, 73247287493299642
Offset: 0
G.f.: A(x) = 1 + 5*x + 29*x^2 + 186*x^3 + 1281*x^4 + 9294*x^5 +...;
A(x) = C(x)*F(x)^2/(1 - x*C(x)*F(x)^2 - x*F(x)^3), where
C(x) = 1 + xC(x)^2 is g.f. of Catalan numbers (A000108):
[1, 1, 2, 5, 14, 42, 132, 429, 1430, ..., C(2n,n)/(n+1), ...] and
F(x) = 1 + xF(x)^4 is g.f. of A002293:
[1, 1, 4, 22, 140, 969, 7084, 53820, ..., C(4n,n)/(3n+1), ...].
-
{a(n)=local(m=n+1,C,F,A); C=Ser(vector(m,r,binomial(2*r-2,r-1)/r)); F=Ser(vector(m,r,binomial(4*r-4,r-1)/(3*r-2))); A=C*F^2/(1-x*C*F^2-x*F^3);polcoeff(A+O(x^m),n,x)}
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