1,2

a(n,m) = R_n^m(a=0,b=2) in the notation of the given reference.

Exponential Riordan array [1/(1+2x),ln(1+2x)/2]. The unsigned triangle is [1/(1-2x),ln(1/sqrt(1-2x))]. [From Paul Barry (pbarry(AT)wit.ie), Apr 29 2009]

Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

W. Lang, First 9 rows and comment.

a(n, m) = a(n-1, m-1) - 2*(n-1)*a(n-1, m), n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0, a(1, 1)=1. E.g.f. for m-th column of signed triangle: (((ln(1+2*x))/2)^m)/m!.

E.g.f.: (1+2*x)^(y/2). O.g.f. for n-th row of signed triangle: Sum_{m=0..n} stirling1(n, m)*2^(n-m)*x^m. - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 11 2003

a(n, m) = S1(n, m)*2^(n-m), with S1(n, m) := A008275(n, m) (signed Stirling1 triangle).

{1}, {2,1}, {8,6,1}, {48,44,12,1}, ...

Contribution from Paul Barry (pbarry(AT)wit.ie), Apr 29 2009: (Start)

The unsigned triangle [1/(1-2x),ln(1/sqrt(1-2x))] has production matrix

2, 1,

4, 4, 1,

8, 12, 6, 1,

16, 32, 24, 8, 1,

32, 80, 80, 40, 10, 1,

64, 192, 240, 160, 60, 12, 1

which is A007318^{2} beheaded. (End)

Table[ Rest@ CoefficientList[ Product[ z-k, {k, 0, 2p-2, 2} ], z ], {p, 6} ]

First column (unsigned triangle) is (2(n-1))!! = 1, 2, 8, 48, 384...= A000165(n-1) and the row sums (unsigned) are (2n-1)!! = 1, 3, 15, 105, 945... = A001147(n-1). Cf. A051141, A051142.

Sequence in context: A008517 A142336 A114193 this_sequence A108084 A108085 A154897

Adjacent sequences: A039680 A039681 A039682 this_sequence A039684 A039685 A039686

sign,tabl

wouter.meeussen(AT)pandora.be

Additional comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de).