%I #46 Sep 30 2019 07:16:12

%S 1,1,0,2,2,0,5,14,5,0,14,74,76,14,0,42,352,698,378,42,0,132,1588,5088,

%T 5404,1808,132,0,429,6946,32461,56410,37546,8484,429,0,1430,29786,

%U 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

%N 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.

%C More generally, we have the following related identity.

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

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

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

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

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

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

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

%H Paul D. Hanna, <a href="/A277295/b277295.txt">Table of n, a(n) for n = 1..1275, of rows n=1..50 of this triangle in flattened form.</a>

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

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

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

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

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

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

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

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

%e 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 +...

%e such that

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

%e Also,

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

%e ...

%e This triangle of coefficients T(n,k) of x^n*y^k in g.f. A(x,y) begins:

%e 1;

%e 1, 0;

%e 2, 2, 0;

%e 5, 14, 5, 0;

%e 14, 74, 76, 14, 0;

%e 42, 352, 698, 378, 42, 0;

%e 132, 1588, 5088, 5404, 1808, 132, 0;

%e 429, 6946, 32461, 56410, 37546, 8484, 429, 0;

%e 1430, 29786, 189940, 486550, 535410, 244220, 39446, 1430, 0;

%e 4862, 126008, 1046190, 3690410, 6036632, 4597402, 1522466, 182732, 4862, 0;

%e 16796, 527900, 5511440, 25518020, 57890956, 66031704, 36873036, 9227504, 846248, 16796, 0;

%e 58786, 2195580, 28061890, 164565240, 493085566, 784844330, 661152388, 281873618, 54885974, 3926338, 58786, 0; ...

%e RELATED SEQUENCES.

%e Given T(n,k) is the coefficient of x^n*y^k in g.f. A(x,y),

%e if b(n) = Sum_{k=0..n-1} T(n,k) * p^k * q^(n-k-1)

%e then B(x) = Sum_{n>=1} b(n)*x^n satisfies

%e (1) B(x - p*B(x)^2) = x + (q-p)*B(x)^2

%e (2) B(x) = x + B( p*B(x) + (q-p)*x )^2.

%e Examples:

%e A213591(n) = sum(k=0,n-1, T(n,k) )

%e A275765(n) = sum(k=0,n-1, T(n,k) * 2^(n-k) )

%e A276360(n) = sum(k=0,n-1, T(n,k) * 3^(n-k-1) )

%e A276361(n) = sum(k=0,n-1, T(n,k) * 2^k * 3^(n-k-1) )

%e A276362(n) = sum(k=0,n-1, T(n,k) * 4^(n-k-1) )

%e A276363(n) = sum(k=0,n-1, T(n,k) * 3^k * 4^(n-k-1) )

%e A276365(n) = sum(k=0,n-1, T(n,k) * 2^k )

%e A277300(n) = sum(k=0,n-1, T(n,k) * 5^(n-k-1) )

%e A277301(n) = sum(k=0,n-1, T(n,k) * 2^k * 5^(n-k-1) )

%e A277302(n) = sum(k=0,n-1, T(n,k) * 3^k * 5^(n-k-1) )

%e A277303(n) = sum(k=0,n-1, T(n,k) * 4^k * 5^(n-k-1) )

%e A277304(n) = sum(k=0,n-1, T(n,k) * 6^(n-k-1) )

%e A277305(n) = sum(k=0,n-1, T(n,k) * 5^k * 6^(n-k-1) )

%e A277306(n) = sum(k=0,n-1, T(n,k) * (-1)^k )

%e A277307(n) = sum(k=0,n-1, T(n,k) * 3^k )

%e A277308(n) = sum(k=0,n-1, T(n,k) * 3^k * 2^(n-k-1) )

%e A277309(n) = sum(k=0,n-1, T(n,k) * 5^k * 2^(n-k-1) )

%e A277310(n) = sum(k=0,n-1, T(n,k) * 4^k )

%e A277311(n) = sum(k=0,n-1, T(n,k) * 5^k )

%e ...

%t 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 T[n_, k_] := SeriesCoefficient[c[n], {y, 0, k}];

%t Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Sep 30 2019 *)

%o (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)}

%o for(n=1, 12, for(k=0,n-1, print1(T(n,k), ", "));print(""))

%Y 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)).

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

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

%K nonn,tabl

%O 1,4

%A _Paul D. Hanna_, Oct 11 2016