0,6

Let R(m,n,k), 0<=k<=n, the Riordan array (1, x*g(x)) where g(x) is g.f. of the m-fold factorials . Then Sum_{k, 0<=k<=n} = R(m,n,k) = Sum_{k, 0<=k<=n} T(n,k)*m^(n-k).

For m = -1, R(-1,n,k) is A026729(n,k).

For m = 0, R(0,n,k) is A097805(n,k).

For m = 1, R(1,n,k) is A084938(n,k).

For m = 2, R(2,n,k) is A111106(n,k).

Sum_{k, 0<=k<=n} (-1)^(n-k)*T(n, k) = A000045(n+1), Fibonacci numbers.

Sum_{k, 0<=k<=n} T(n, k) = A051295(n).

Sum_{k, 0<=k<=n} 2^(n-k)*T(n, k) = A112934(n).

T(0, 0) = 1, T(n, n) = 2^(n-1).

G.f.: A(x, y) = 1/(1 - x*y*Sum_{j>=0} (y-1+j)!/(y-1)!*x^j ). - Paul D. Hanna (pauldhanna(AT)juno.com), Oct 26 2005

Triangle begins:

.1;

.0, 1;

.0, 0, 2;

.0, 0, 1, 4;

.0, 0, 2, 5, 8;

.0, 0, 6, 15, 17, 16;

.0, 0, 24, 62, 68, 49, 32;

.0, 0, 120, 322, 359, 243, 129, 64;

.0, 0, 720, 2004, 2308, 1553, 756, 321, 128;

.0, 0, 5040, 14508, 17332, 11903, 5622, 2151, 769, 256;

.0, 0, 40320, 119664, 148232, 105048, 49840, 18066, 5756, 1793, 512;

....................................................................

At y=2: Sum_{k=0..n} 2^k*T(n,k) = A113327(n) where (1 + 2*x + 8*x^2 + 36*x^3 +...+ A113327(n)*x^n +..) = 1/(1 - 2/1!*x*(1! + 2!*x + 3!*x^2 + 4!*x^3 +..) ).

At y=3: Sum_{k=0..n} 3^k*T(n,k) = A113328(n) where (1 + 3*x + 18*x^2 + 117*x^3 +...+ A113328(n)*x^n +..) = 1/(1 - 3/2!*x*(2! + 3!*x + 4!*x^2 + 5!*x^3 +..) ).

At y=4: Sum_{k=0..n} 4^k*T(n,k) = A113329(n) where (1 + 4*x + 32*x^2 + 272*x^3 +...+ A113329(n)*x^n +..) = 1/(1 - 4/3!*x*(3! + 4!*x + 5!*x^2 + 6!*x^3 +..) ).

(PARI) {T(n, k)=local(x=X+X*O(X^n), y=Y+Y*O(Y^k)); A=1/(1-x*y*sum(j=0, n, x^j*prod(i=0, j-1, y+i))); return(polcoeff(polcoeff(A, n, X), k, Y))} (Hanna)

Cf. A000045, A026729, A051295, A084938, A097805, A111106, A112934.

Cf. m-fold factorials : A000142, A001147, A007559, A007696, A008548, A008542.

Cf. A113326, A113327 (y=2), A113328 (y=3), A113329 (y=4), A113330 (y=5), A113331 (y=6).

Sequence in context: A123634 A091866 A168511 this_sequence A109077 A137585 A072458

Adjacent sequences: A111143 A111144 A111145 this_sequence A111147 A111148 A111149

easy,nonn,tabl

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 19 2005