rwg>This biwedge extension is typical of many number triangles, including Stirling and Eulerian. Had to switch to Macsyma--Mma remains extremely stodgy in these matters. Binomials: (c30) PT(N,FRM) := GENMATRIX(LAMBDA([I,J],SUBST("",0,IF EVENP(I+J) THEN 0 ELSE BINOMIAL(I-1,(I+J+1)/2-FLOOR(N/4)))),N,N,FRM,FRM)$ (c31) PT(4,-3); [ 1 ] [ ] [ 1 ] [ ] [ 1 - 2 ] [ ] [ 1 - 1 ] (d31) [ ] [ 1 ] [ ] [ 1 1 ] [ ] [ 1 2 1 ] [ ] [ 1 3 3 1 ] StirlingS1: (c32) S1T(N,FRM) := GENMATRIX(LAMBDA([I,J],SUBST("",0,IF EVENP(I+J) THEN 0 ELSE STIRLING_S1(I-1,(I+J+1)/2-FLOOR(N/4)))),N,N,FRM,FRM)$ (c33) S1T(5,-7); [ ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 15 1 ] [ ] [ 10 1 ] [ ] [ 25 6 1 ] [ ] (d33) [ 7 3 1 ] [ ] [ 1 1 1 1 ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 1 1 ] [ ] [ 2 3 1 ] [ ] [ 6 11 6 1 ] Mirror image due to argument conventions. StirlingS2: (c34) S2T(N,FRM) := GENMATRIX(LAMBDA([I,J],SUBST("",0,IF EVENP(I+J) THEN 0 ELSE STIRLING_S2(I-1,(I+J+1)/2-FLOOR(N/4)))),N,N,FRM,FRM)$ (c35) S2T(5,-7); [ ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 15 1 ] [ ] [ 10 1 ] [ ] [ 35 6 1 ] [ ] (d35) [ 11 3 1 ] [ ] [ 6 2 1 1 ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 1 1 ] [ ] [ 1 3 1 ] [ ] [ 1 7 6 1 ] Eulerian: (c36) ET(N,FRM) := GENMATRIX(LAMBDA([I,J],SUBST("",0,IF EVENP(I+J) THEN 0 ELSE EULERIAN(I-1,(I+J+1)/2-FLOOR(N/4)))),N,N,FRM,FRM)$ (c37) ET(5,-5); [ ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 17 ] [ 1 -- ] [ 8 ] [ ] [ 5 ] [ 1 - ] [ 4 ] (d37) [ ] [ 1 1 ] [ 1 - - ] [ 2 3 ] [ ] [ 1 ] [ ] [ 1 ] [ ] [ 1 1 ] [ ] [ 1 4 1 ] [ ] [ 1 11 11 1 ] See p8 of http://go.helms-net.de/math/binomial_new/01_12_Eulermatrix.pdf --rwg