A089949 Triangle T(n,k), read by rows, given by [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 12, 34, 24, 0, 1, 20, 110, 210, 120, 0, 1, 30, 270, 974, 1452, 720, 0, 1, 42, 560, 3248, 8946, 11256, 5040, 0, 1, 56, 1036, 8792, 38338, 87504, 97296, 40320, 0, 1, 72, 1764, 20580, 129834, 463050, 920184, 930960, 362880

Offset

0,6

Comments

Row reverse appears to be A111184. - Peter Bala, Feb 17 2017

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

Sum_{k=0..n} x^(n-k)*T(n,k) = A111528(x, n); see A000142, A003319, A111529, A111530, A111531, A111532, A111533 for x = 0, 1, 2, 3, 4, 5, 6. - Philippe Deléham, Aug 09 2005

Sum_{k=0..n} T(n,k)*3^k = A107716(n). - Philippe Deléham, Aug 15 2005

Sum_{k=0..n} T(n,k)*2^k = A000698(n+1). - Philippe Deléham, Aug 15 2005

G.f.: A(x, y) = (1/x)*(1 - 1/(1 + Sum_{n>=1} [Product_{k=0..n-1}(1+k*y)]*x^n )). - Paul D. Hanna, Aug 16 2005

Example

Triangle begins:
  1;
  0, 1;
  0, 1,  2;
  0, 1,  6,   6;
  0, 1, 12,  34,  24;
  0, 1, 20, 110, 210,  120;
  0, 1, 30, 270, 974, 1452, 720; ...

Mathematica

m = 10;
gf = (1/x)*(1-1/(1+Sum[Product[(1+k*y), {k, 0, n-1}]*x^n, {n, 1, m}]));
CoefficientList[#, y]& /@ CoefficientList[gf + O[x]^m, x] // Flatten (* Jean-François Alcover, May 11 2019 *)

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==0, 1, if(k==0, 0, polcoeff(polcoeff( (1-1/(1+sum(m=1, n+k, prod(j=0, m-1, 1+j*y)*x^m)))/x +x*O(x^n), n, x)+y*O(y^k), k, y)))) \\ Paul D. Hanna, Aug 16 2005

Crossrefs

Cf. A084938, A111184.

Diagonals: A000007, A000012, A002378, A000142.

Row sums: A003319.

Sequence in context: A293147 A331047 A264550 * A085845 A138106 A131689

Adjacent sequences: A089946 A089947 A089948 * A089950 A089951 A089952

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Jan 11 2004

Status

approved