A116077 Triangular matrix T, read by rows, such that the anticommutator of T and U shifts the columns of T up 1 row: {T,U}(n,k) = T(n+1,k), where U denotes the triangular matrix defined by U(n,k) = A000108(n-k) = Catalan(n-k) for n>=k and where T(n,n) = (n+1).

1, 2, 2, 7, 4, 3, 28, 13, 6, 4, 117, 48, 19, 8, 5, 496, 187, 68, 25, 10, 6, 2110, 748, 257, 88, 31, 12, 7, 8968, 3034, 1000, 327, 108, 37, 14, 8, 38017, 12400, 3958, 1252, 397, 128, 43, 16, 9, 160648, 50887, 15832, 4882, 1504, 467, 148, 49, 18, 10

Offset

0,2

Comments

Difference between any two columns = A000984 (central binomial coefficients):

T(n+1,k+1) = T(n,k) + A000984(n-k).

Links

Table of n, a(n) for n=0..54.

Formula

G.f. of column k: k/sqrt(1-4*x) + 4*x^2/(1+sqrt(1-4*x))^2/(1-4*x)^(3/2).

G.f.: A(x,y) = 1/sqrt(1-4*x)/(1-x*y)^2 + 4*x^2/(1+sqrt(1-4*x))^2/(1-4*x)^(3/2)/(1-x*y).

Example

Triangular matrix T begins:
1;
2,2;
7,4,3;
28,13,6,4;
117,48,19,8,5;
496,187,68,25,10,6;
2110,748,257,88,31,12,7;
8968,3034,1000,327,108,37,14,8;
38017,12400,3958,1252,397,128,43,16,9; ...
Define triangular matrix U by U(n,k) = Catalan(n-k):
1;
1,1;
2,1,1;
5,2,1,1;
14,5,2,1,1; ...
Then anticommutator {T,U} = T*U + U*T is given by
{T,U}(n,k) = T(n+1,k):
2;
7,4;
28,13,6;
117,48,19,8;
496,187,68,25,10; ...
which equals T with columns shift up 1 row.
The commutator [T,U] = T*U - U*T is given by
[T,U](n,k) = 4^(n-k) - C(2*(n-k)+1,n-k):
0,
1,0,
6,1,0,
29,6,1,0,
130,29,6,1,0, ...
Further, commutator [U^-1,T] = (U^-1)*T - T*(U^-1) is
[U^-1,T](n,k) = 4^(n-k-1) if n>k, else 0:
0;
1,0;
4,1,0;
16,4,1,0;
64,16,4,1,0; ...

Mathematica

T[n_, k_] := SeriesCoefficient[1/Sqrt[1 - 4 x]/(1 - x y)^2 + 4 x^2/(1 + Sqrt[1 - 4 x])^2/(1 - 4 x)^(3/2)/(1 - x y), {x, 0, n}, {y, 0, k}];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 24 2019 *)

Prog

(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff( 1/(1-X*Y)^2/sqrt(1-4*X)+4*X^2/(1+sqrt(1-4*X))^2/(1-4*X)^(3/2)/(1-X*Y), n, x), k, y)}

Crossrefs

Cf. A116078 (column 0), A000108, A000984.

Sequence in context: A245286 A075428 A277199 * A074144 A239383 A325525

Adjacent sequences: A116074 A116075 A116076 * A116078 A116079 A116080

Keyword

nonn,tabl

Author

Paul D. Hanna, Feb 04 2006

Status

approved