0,4

Row sums are A000165. [From Paul Barry (pbarry(AT)wit.ie), Feb 07 2009]

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

Triangle T(n, k), read by rows, given by [1, 2, 3, 4, 5, 6, 7, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...] where DELTA is the operator defined in A084938 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 20 2005

T(n, k) = sum[(-2)^(n-i) binomial(i, k) s(n, i), i=k..n] where s(n, k) are signed Stirling numbers of the first kind. - Francis Woodhouse (fwoodhouse(AT)gmail.com), Nov 18 2005

G.f.: G.f.: 1/(1-(x+xy)/(1-2x/(1-(3x+xy)/(1-4x/(1-(5x+xy)/(1-6xy/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Feb 07 2009]

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009: (Start)

a(n, m) = (2*n-1)*a(n-1,m)+a(n-1,m-1) with a(n, 0) = (2*n-1)!! and a(n, n) = 1.

(End)

E.g. For n=4, (x + 1) (x + 3) (x + 5) (x + 7) = x^4+16*x^3+86*x^2+176*x+105

1; 1,1; 3,4,1; 15,23,9,1; 105,176,86,16,1; ...

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009: (Start)

nmax:=8; mmax:=nmax: for n from 0 to nmax do a(n, 0):=doublefactorial(2*n-1) od: for n from 0 to nmax do a(n, n):=1 od: for n from 2 to nmax do for m from 1 to n-1 do a(n, m) := (2*n-1)*a(n-1, m)+a(n-1, m-1) od; od: T:=0; for n from 0 to nmax do for m from 0 to n do a(T):=a(n, m); T:=T+1 od: od: seq(a(n), n=0..T-1);

(End)

T[n_, k_] := Sum[(-2)^(n-i) Binomial[i, k] StirlingS1[n, i], {i, k, n}] (Woodhouse)

A039757 is signed version.

Diagonals : A001147, A004041, A028339, A028340, A028341; A000012, A000290, A024196, A024197, A024198. Row sums : A000165

Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Jun 08 2009: (Start)

A161198 is a scaled triangle version and A109692 is a transposed triangle version.

(End)

Sequence in context: A114189 A059110 A100326 this_sequence A039757 A136228 A154829

Adjacent sequences: A028335 A028336 A028337 this_sequence A028339 A028340 A028341

tabl,nonn,easy,nice

R. W. Gosper