A277295 G.f. A(x,y) satisfies: A( x - y*A(x,y)^2, y) = x + (1-y)*A(x,y)^2, where the coefficients T(n,k) of x^n*y^k form a triangle read by rows n>=1, for k=0..n-1.

1, 1, 0, 2, 2, 0, 5, 14, 5, 0, 14, 74, 76, 14, 0, 42, 352, 698, 378, 42, 0, 132, 1588, 5088, 5404, 1808, 132, 0, 429, 6946, 32461, 56410, 37546, 8484, 429, 0, 1430, 29786, 189940, 486550, 535410, 244220, 39446, 1430, 0, 4862, 126008, 1046190, 3690410, 6036632, 4597402, 1522466, 182732, 4862, 0, 16796, 527900, 5511440, 25518020, 57890956, 66031704, 36873036, 9227504, 846248, 16796, 0, 58786, 2195580, 28061890, 164565240, 493085566, 784844330, 661152388, 281873618, 54885974, 3926338, 58786, 0

Offset

1,4

Comments

More generally, we have the following related identity.

Given functions F and G with F(0)=0, F'(0)=1, G(0)=0, G'(0)=0,

if F(x - y*G(x)) = x + (1-y)*G(x), then

(1) F(x) = x + G( y*F(x) + (1-y)*x ),

(2) y*F(x) + (1-y)*x = Series_Reversion(x - y*G(x)),

(3) F(x) = x + G(x + y*G(x + y*G(x + y*G(x +...)))),

(4) F(x) = x + Sum_{n>=1} y^(n-1) * d^(n-1)/dx^(n-1) G(x)^n / n!.

The g.f. of this sequence A(x,y) equals F(x) in the above when G(x) = F(x)^2.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1275, of rows n=1..50 of this triangle in flattened form.

Formula

G.f. A(x,y) also satisfies:

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

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

(3) y*x + (1-y)*B(x,y) = Series_Reversion( x + (1-y)*A(x,y)^2 ), where B( A(x,y), y) = x.

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

In formulas 2 and 3, the series reversion is taken with respect to variable x.

T(n+1,0) = T(n+1,n-1) = binomial(2*n,n)/(n+1) = A000108(n) for n>=1.

T(n+1,1) = 4^n - (3*n+1)*binomial(2*n,n)/(n+1) = A138156(n-1) for n>=1.

Example

G.f.: A(x,y)  = x + x^2 + (2*y + 2)*x^3 + (5*y^2 + 14*y + 5)*x^4 + (14*y^3 + 76*y^2 + 74*y + 14)*x^5 + (42*y^4 + 378*y^3 + 698*y^2 + 352*y + 42)*x^6 + (132*y^5 + 1808*y^4 + 5404*y^3 + 5088*y^2 + 1588*y + 132)*x^7 + (429*y^6 + 8484*y^5 + 37546*y^4 + 56410*y^3 + 32461*y^2 + 6946*y + 429)*x^8 + (1430*y^7 + 39446*y^6 + 244220*y^5 + 535410*y^4 + 486550*y^3 + 189940*y^2 + 29786*y + 1430)*x^9 + (4862*y^8 + 182732*y^7 + 1522466*y^6 + 4597402*y^5 + 6036632*y^4 + 3690410*y^3 + 1046190*y^2 + 126008*y + 4862)*x^10 +...
such that
A( x - y*A(x,y)^2, y)  =  x + (1-y)*A(x,y)^2.
Also,
A(x,y) = x + A( y*A(x,y) + (1-y)*x, y)^2.
...
This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins:
1;
1, 0;
2, 2, 0;
5, 14, 5, 0;
14, 74, 76, 14, 0;
42, 352, 698, 378, 42, 0;
132, 1588, 5088, 5404, 1808, 132, 0;
429, 6946, 32461, 56410, 37546, 8484, 429, 0;
1430, 29786, 189940, 486550, 535410, 244220, 39446, 1430, 0;
4862, 126008, 1046190, 3690410, 6036632, 4597402, 1522466, 182732, 4862, 0;
16796, 527900, 5511440, 25518020, 57890956, 66031704, 36873036, 9227504, 846248, 16796, 0;
58786, 2195580, 28061890, 164565240, 493085566, 784844330, 661152388, 281873618, 54885974, 3926338, 58786, 0; ...
RELATED SEQUENCES.
Given T(n,k) is the coefficient of x^n*y^k in g.f. A(x,y),
if b(n) = Sum_{k=0..n-1} T(n,k) * p^k * q^(n-k-1)
then B(x) = Sum_{n>=1} b(n)*x^n satisfies
(1) B(x - p*B(x)^2) = x + (q-p)*B(x)^2
(2) B(x)  =  x + B( p*B(x) + (q-p)*x )^2.
Examples:
A213591(n) = sum(k=0,n-1, T(n,k) )
A275765(n) = sum(k=0,n-1, T(n,k) * 2^(n-k) )
A276360(n) = sum(k=0,n-1, T(n,k) * 3^(n-k-1) )
A276361(n) = sum(k=0,n-1, T(n,k) * 2^k * 3^(n-k-1) )
A276362(n) = sum(k=0,n-1, T(n,k) * 4^(n-k-1) )
A276363(n) = sum(k=0,n-1, T(n,k) * 3^k * 4^(n-k-1) )
A276365(n) = sum(k=0,n-1, T(n,k) * 2^k )
A277300(n) = sum(k=0,n-1, T(n,k) * 5^(n-k-1) )
A277301(n) = sum(k=0,n-1, T(n,k) * 2^k * 5^(n-k-1) )
A277302(n) = sum(k=0,n-1, T(n,k) * 3^k * 5^(n-k-1) )
A277303(n) = sum(k=0,n-1, T(n,k) * 4^k * 5^(n-k-1) )
A277304(n) = sum(k=0,n-1, T(n,k) * 6^(n-k-1) )
A277305(n) = sum(k=0,n-1, T(n,k) * 5^k * 6^(n-k-1) )
A277306(n) = sum(k=0,n-1, T(n,k) * (-1)^k )
A277307(n) = sum(k=0,n-1, T(n,k) * 3^k )
A277308(n) = sum(k=0,n-1, T(n,k) * 3^k * 2^(n-k-1) )
A277309(n) = sum(k=0,n-1, T(n,k) * 5^k * 2^(n-k-1) )
A277310(n) = sum(k=0,n-1, T(n,k) * 4^k )
A277311(n) = sum(k=0,n-1, T(n,k) * 5^k )
...

Mathematica

c[n_] := c[n] = Module[{A}, A[x_] = x; Do[A[x_] = x + A[y A[x] + (1-y) x + x O[x]^j]^2, {j, n}] // Normal; SeriesCoefficient[A[x], {x, 0, n}] // Expand];
T[n_, k_] := SeriesCoefficient[c[n], {y, 0, k}];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Sep 30 2019 *)

Prog

(PARI) {T(n, k) = my(A=x); for(i=1, n, A = x + subst(A^2, x, y*A + (1-y)*x +x*O(x^n)) ); polcoeff(polcoeff(A, n, x), k, y)}
for(n=1, 12, for(k=0, n-1, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A000108 (column 0), A138156 (column 1), A277296 (column 2), A277297 (diagonal), A277298 (central terms T(2*n-1,n-1)), A277299 (central terms T(2*n,n-1)).

Cf. A213591 (row sums), A275765, A276360, A276361, A276362, A276363, A276365.

Cf. A277300, A277302, A277303, A277304, A277305, A277306, A277307, A277308, A277309, A277310, A277311.

Sequence in context: A355570 A168505 A100334 * A254749 A129936 A253180

Adjacent sequences: A277292 A277293 A277294 * A277296 A277297 A277298

Keyword

nonn,tabl

Author

Paul D. Hanna, Oct 11 2016

Status

approved