0,4

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

Reversal of A094638 .

Equals A132393*A007318, as infinite lower triangular matrices . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 13 2007

T. D. Noe, Rows n=0..50 of triangle, flattened

T(0,0)=1, T(k,n)=0 if k>n or if n<0, T(n,k)=T(n-1,k-1)+n*T(n-1,k). T(n,0)=n!=A000142(n). T(2*n,n)=A129505(n+1). Sum_{k, 0<=k<=n}T(n,k)=(n+1)!=A000142(n+1). Sum_{k, 0<=k<=n}T(n,k)^2=A047796(n+1). T(n,k)=|Stirling1(n+1,k+1)|, see A008275 . (x+1)(x+2)...(x+n)=Sum_{k, 0<=k<=n}T(n,k)*x^n.

Sum_{k, 0<=k<=n}T(n,k)*x^k = A000007(n), A000142(n), A000142(n+1), A001710(n+2), A001715(n+3), A001720(n+4), A001725(n+5), A001730(n+6), A049388(n), A049389(n), A049398(n), A051431(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 13 2007

Triangle begins:

1;

1, 1;

2, 3, 1;

6, 11, 6, 1;

24, 50, 35, 10, 1;

120, 274, 225, 85, 15, 1;

720, 1764, 1624, 735, 175, 21, 1;

5040, 13068, 13132, 6769, 1960, 322, 28, 1;

40320, 109584, 118124, 67284, 22449, 4536, 546, 36, 1;

362880, 1026576, 1172700, 723680, 269325, 63273, 9450, 870, 45, 1 ;...

Diagonals : A000012 A000217 A000914 A001303 A000915 A053567 A112002. Columns A000142 A000254 A000399 A000454 A000482 A001233 A001234.

Adjacent sequences: A130531 A130532 A130533 this_sequence A130535 A130536 A130537

Sequence in context: A138771 A121748 A008275 this_sequence A107416 A105613 A135894

nonn,tabl

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 09 2007