A365770 Expansion of g.f. A(x,y) satisfying A(x,y) = 1 + x*A(x,y)/(1 - x*y * A(x,y))^2, as a triangle of coefficients T(n,k) of x^n*y^k in A(x,y), read by rows n >= 0.

1, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 12, 20, 4, 0, 1, 20, 70, 50, 5, 0, 1, 30, 180, 280, 105, 6, 0, 1, 42, 385, 1050, 882, 196, 7, 0, 1, 56, 728, 3080, 4620, 2352, 336, 8, 0, 1, 72, 1260, 7644, 18018, 16632, 5544, 540, 9, 0, 1, 90, 2040, 16800, 57330, 84084, 51480, 11880, 825, 10, 0, 1, 110, 3135, 33660, 157080, 336336, 330330, 141570, 23595, 1210, 11, 0

Offset: 0

Author

Paul D. Hanna, Oct 10 2023

Keywords

• nonn
• tabl

Comments

A365771(n) = T(2*n,n), the central terms.

A109081(n) = Sum_{k=0..n} T(n,k), the row sums.

A365772(n) = Sum_{k=0..n} T(n,k) * 2^k.

A365773(n) = Sum_{k=0..n} T(n,k) * 3^k.

A365774(n) = Sum_{k=0..n} T(n,k) * 4^k.

A365775(n) = Sum_{k=0..n} T(n,k) * 5^k.

Related identities which hold formally for all Maclaurin series F(x):

(1) F(x) = (1/x) * Sum{n>=1} n^(n-1) * x^n * F(x)^n / (1 + n*x*F(x))^(n+1),

(2) F(x) = (2/x) * Sum{n>=1} n*(n+1)^(n-2) * x^n * F(x)^n / (1 + (n+1)*x*F(x))^(n+1),

(3) F(x) = (3/x) * Sum{n>=1} n*(n+2)^(n-2) * x^n * F(x)^n / (1 + (n+2)*x*F(x))^(n+1),

(4) F(x) = (4/x) * Sum{n>=1} n*(n+3)^(n-2) * x^n * F(x)^n / (1 + (n+3)*x*F(x))^(n+1),

(5) F(x) = (k/x) * Sum{n>=1} n*(n+k-1)^(n-2) * x^n * F(x)^n / (1 + (n+k-1)*x*F(x))^(n+1) for all fixed nonzero k.

Examples

			G.f.: A(x,y) = 1 + x + (1 + 2*y)*x^2 + (1 + 6*y + 3*y^2)*x^3 + (1 + 12*y + 20*y^2 + 4*y^3)*x^4 + (1 + 20*y + 70*y^2 + 50*y^3 + 5*y^4)*x^5 + (1 + 30*y + 180*y^2 + 280*y^3 + 105*y^4 + 6*y^5)*x^6 + (1 + 42*y + 385*y^2 + 1050*y^3 + 882*y^4 + 196*y^5 + 7*y^6)*x^7 + (1 + 56*y + 728*y^2 + 3080*y^3 + 4620*y^4 + 2352*y^5 + 336*y^6 + 8*y^7)*x^8 + ...
where
A(x,y) = 1 + x*A(x,y)/(1 - x*y*A(x,y))^2.
Also,
A(x,y) = 1 + 1^0*x*A(x,y)/(1 + (1-y)*x*A(x,y))^2 + 2^1*x^2*A(x,y)^2/(1 + (2-y)*x*A(x,y))^3 + 3^2*x^3*A(x,y)^3/(1 + (3-y)*x*A(x,y))^4 + 4^3*x^4*A(x,y)^4/(1 + (4-y)*x*A(x,y))^5 + 5^4*x^5*A(x,y)^5/(1 + (5-y)*x*A(x,y))^6 + ...
and
A(x,y) = 1 + (1+y)*1*(1+y)^(-1)*x*A(x,y)/(1 + 1*x*A(x,y))^2 + (1+y)*2*(2+y)^0*x^2*A(x,y)^2/(1 + 2*x*A(x,y))^3 + (1+y)*3*(3+y)^1*x^3*A(x,y)^3/(1 + 3*x*A(x,y))^4 + (1+y)*4*(4+y)^2*x^4*A(x,y)^4/(1 + 4*x*A(x,y))^5 + ...
This triangle of coefficients of x^n*y^k in A(x,y) begins:
1;
1, 0;
1, 2, 0;
1, 6, 3, 0;
1, 12, 20, 4, 0;
1, 20, 70, 50, 5, 0;
1, 30, 180, 280, 105, 6, 0;
1, 42, 385, 1050, 882, 196, 7, 0;
1, 56, 728, 3080, 4620, 2352, 336, 8, 0;
1, 72, 1260, 7644, 18018, 16632, 5544, 540, 9, 0;
1, 90, 2040, 16800, 57330, 84084, 51480, 11880, 825, 10, 0; ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1325

Crossrefs

Cf. A109081 (y=1), A365772 (y=2), A365773 (y=3), A365774 (y=4), A365775 (y=5).

Cf. A365771 (central terms).

Programs

PARI

{T(n,k) = binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k)}
for(n=0,10, for(k=0,n, print1(T(n,k),", "));print(""))

Formula

T(n,k) = binomial(n+1, n-k)/(n+1) * binomial(2*n-k-1, k).

G.f. A(x,y) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*x^n*y^k satisfies the following formulas.

(1) A(x,y) = 1 + x*A(x,y)/(1 - x*y*A(x,y))^2.

(2) A(x,y) = (1/x) * Series_Reversion( x/(1 + x/(1 - x*y)^2) ), where reversion is taken wrt x.

(3) A( x/(1 + x/(1 - x*y)^2), y) = 1 + x/(1 - x*y)^2.

(4) A(x,y) = 1 + (1+y) * Sum{n>=1} n*(n+y)^(n-2) * x^n * A(x,y)^n / (1 + n*x*A(x,y))^(n+1).

(5) A(x,y) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n * A(x,y)^n / (1 + (n+m-y)*x*A(x,y))^(n+1) for all fixed nonnegative m.

(5.a) A(x,y) = 1 + Sum{n>=1} n^(n-1) * x^n * A(x,y)^n / (1 + (n-y)*x*A(x,y))^(n+1).

(5.b) A(x,y) = 1 + 2 * Sum{n>=1} n*(n+1)^(n-2) * x^n * A(x,y)^n / (1 + (n+1-y)*x*A(x,y))^(n+1).

(5.c) A(x,y) = 1 + 3 * Sum{n>=1} n*(n+2)^(n-2) * x^n * A(x,y)^n / (1 + (n+2-y)*x*A(x,y))^(n+1).

(5.d) A(x,y) = 1 + 4 * Sum{n>=1} n*(n+3)^(n-2) * x^n * A(x,y)^n / (1 + (n+3-y)*x*A(x,y))^(n+1).