1,5

A convolution triangle of numbers obtained from A019590.

a(n,m) := s1(-1; n,m), a member of a sequence of triangles including s1(0; n,m)= A023531(n,m) (unit matrix) and s1(2; n,m)= A007318(n-1,m-1) (Pascal's triangle).

The signed triangular matrix a(n,m)*(-1)^(n-m) is the inverse matrix of the triangular Catalan convolution matrix A033184(n+1,m+1), n >= m >= 0, with A033184(n,m) := 0 if n<m.

Riordan array (1+x, x(1+x)). The signed triangle is the Riordan array (1-x,x(1-x)), inverse to (c(x),xc(x)) with c(x) g.f. for A000108. - Paul Barry (pbarry(AT)wit.ie), Feb 02 2005

Also, a(n,k)=number of compositions of n into k parts of 1's and 2's. Example: a(6,4)=6 because we have 2211, 2121, 2112, 1221, 1212, and 1122. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 05 2005

Subtriangle of A026729 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 31 2006

W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.

D. Merlini, R. Sprugnoli and M. C. Verri, An algebra for proper generating trees

a(n, m) = 2*(2*m-n+1)*a(n-1, m)/n + m*a(n-1, m-1)/n, n >= m >= 1; a(n, m) := 0, n<m; a(n, 0) := 0; a(1, 1)=1. G.f. for m-th column: (x*(1+x))^m.

As a number triangle with offset 0, this is T(n, k)=sum{k=0..n, (-1)^(n+i)C(n, i)C(i+k+1, 2k+1)}. The diagonal sums give the Padovan sequence A000931(n+5). Inverse binomial transform of A078812 (product of lower triangular matrices). - Paul Barry (pbarry(AT)wit.ie), Jun 21 2004

{1}; {1,1}; {0,2,1}; {0,1,3,1}; {0,0,3,4,1}; ...

for n from 1 to 12 do seq(binomial(k, n-k), k=1..n) od; # yields sequence in triangular form (Deutsch)

Row sums A000045(n+1) (Fibonacci). a(n, 1)= A019590(n) (Fermat's last theorem). Cf. A049403.

Adjacent sequences: A030525 A030526 A030527 this_sequence A030529 A030530 A030531

Sequence in context: A129558 A131185 A052249 this_sequence A077227 A089263 A047265

easy,nonn,tabl

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

More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 05 2005