A105632 - OEIS

%I #19 Jun 12 2023 09:20:42

%S 1,1,1,2,1,1,4,3,1,1,9,7,4,1,1,21,19,10,5,1,1,51,51,31,13,6,1,1,127,

%T 141,91,45,16,7,1,1,323,393,276,141,61,19,8,1,1,835,1107,834,461,201,

%U 79,22,9,1,1,2188,3139,2535,1485,701,271,99,25,10,1,1,5798,8953,7711,4803,2381,1001,351,121,28,11,1,1

%N Triangle, read by rows, where the g.f. A(x,y) satisfies the equation: A(x,y) = 1/(1-x*y) + x*A(x,y) + x^2*A(x,y)^2.

%C Column 0 is A001006 (Motzkin numbers). Column 1 is A002426 (Central trinomial coefficients). Row sums form A105633 (also equal to A057580?).

%C T(n,k) is the number of UUDU-avoiding Dyck paths of semilength n+1 with k UDUs, where U = (1,1) is an upstep and D = (1,-1) is a downstep. For example, T(3,1) = 3 counts UDUUUDDD, UDUUDDUD, UUDDUDUD. - _David Callan_, Nov 25 2021

%F G.f. for column k (k>0): Sum_{j=0..k-1} C(k-1, j)*A000108(j)*x^(2*j)/(1-2*x-3*x^2)^(j+1/2), where A000108(j) = binomial(2*j, j)/(j+1) is the j-th Catalan number.

%F G.f.: A(x, y) = (1-x - sqrt((1-x)^2 - 4*x^2/(1-x*y)))/(2*x^2).

%e Triangle begins:

%e     1;

%e     1,    1;

%e     2,    1,   1;

%e     4,    3,   1,   1;

%e     9,    7,   4,   1,   1;

%e    21,   19,  10,   5,   1,  1;

%e    51,   51,  31,  13,   6,  1,  1;

%e   127,  141,  91,  45,  16,  7,  1, 1;

%e   323,  393, 276, 141,  61, 19,  8, 1, 1;

%e   835, 1107, 834, 461, 201, 79, 22, 9, 1, 1; ...

%e Let G = (1-2*x-3*x^2), then the column g.f.s are:

%e k=1: 1/sqrt(G)

%e k=2: (G + (1)*1*x^2)/sqrt(G^3)

%e k=3: (G^2 + (1)*2*x^2*G + (2)*1*x^4)/sqrt(G^5)

%e k=4: (G^3 + (1)*3*x^2*G^2 + (2)*3*x^4*G + (5)*1*x^6)/sqrt(G^7)

%e k=5: (G^4 + (1)*4*x^2*G^3 + (2)*6*x^4*G^2 + (5)*4*x^6*G + (14)*1*x^8)/sqrt(G^9)

%e and involve Catalan numbers and binomial coefficients.

%e MATRIX INVERSE.

%e The matrix inverse starts

%e      1;

%e     -1,   1;

%e     -1,  -1,   1;

%e      0,  -2,  -1,  1;

%e      2,  -1,  -3, -1,  1;

%e      6,   2,  -2, -4, -1,  1;

%e     13,  10,   2, -3, -5, -1,  1;

%e     18,  32,  14,  2, -4, -6, -1,  1;

%e    -12,  76,  56, 18,  2, -5, -7, -1,  1;

%e   -206, 108, 162, 86, 22,  2, -6, -8, -1, 1;

%e - _R. J. Mathar_, Apr 08 2013

%p A105632 := proc(n,k)

%p     (1-x-sqrt((1-x)^2-4*x^2/(1-x*y)))/2/x^2 ;

%p     coeftayl(%,x=0,n) ;

%p     coeftayl(%,y=0,k) ;

%p end proc: # _R. J. Mathar_, Apr 08 2013

%t T[n_, k_] := SeriesCoefficient[(1 - x - Sqrt[(1 - x)^2 - 4*x^2/(1 - x*y)])/(2*x^2), {x, 0, n}] // SeriesCoefficient[#, {y, 0, k}]&;

%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 10 2023 *)

%o (PARI) {T(n,k)=local(A=1+x+x*y+x*O(x^n)+y*O(y^k)); for(i=1,n,A=1/(1-x*y)+x*A+x^2*A^2);polcoeff(polcoeff(A,n,x),k,y)}

%o (PARI) {T(n,k)=local(X=x+x*O(x^n),Y=y+y*O(y^k));polcoeff(polcoeff( 2/(1-X+sqrt((1-X)^2-4*X^2/(1-X*Y)))/(1-X*Y),n,x),k,y)}

%Y Cf. A105633 (row sums), A001006 (column 0), A002426 (column 1).

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Apr 17 2005