Warren, you have GMail. Neil just taught me to click on▼, then "Message text garbled?" That should fix everything but the overwrapped lines, which you can fix by copying and pasting into an editor and deleting the bogus linebreaks. I too hate ASCII art. Why can't math-fun allow html with in-line images?
From the path-invariant 3x3s {{j, {{0, q^(j + 2*n), 1 - q^n}, {1, q^n, 0}, {0, 0, 1}}}, {n, {{q^(j + 2*n)/(1 - q^(1 + n)), 0, 1 - q^n}, {0, q^(1 + j + 2*n)/(1 - q^(1 + n)),1}, {0, 0, 1}}}}
Neil and I just found [ inf ] [ ==== n n (n + 1) ] [ \ a q ] [ ? a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ j ] [ n = 0 ] | | [ 0 a q ] = [ ] | | [ ] [ inf 2 ] j = 0 [ 1 1 ] [ ==== n n ] [ \ a q ] [ ? > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] Maybe Corey&Julian can help me determine the "?"s. --rwg I only recently learned that a lot of broken links to gosper,org are due to a disk crash. If you hit one, send me mail and I'll try to restore it. On Sun, Oct 7, 2012 at 4:02 PM, Bill Gosper <billgosper@gmail.com> wrote:
(For R-R, special-case a:=q)
(c44) 'PRODUCT(MATRIX([0,A],[Q^K,1]),K,0,INF) = MATRIX([0,SUM(A^(N+1)*Q^N^2/QPOCH(Q,Q,N),N,0,INF)],[0,SUM(A^N*Q^(N^2-N)/QPOCH(Q,Q,N),N,0,INF)]);
[ inf 2 ] [ ==== n n ] [ \ a q ] [ 0 a > -------------- ] inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 a ] [ n = 0 ] (d44) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 0 [ q 1 ] [ ==== n n - n ] [ \ a q ] [ 0 > -------------- ] [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ]
(c45) TAYLOR(PRUD(PART(%,1,1),K,0,7) = MAKEPROD(RHS(%)),Q,0,6);
[ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ] (d45)/T/ [ ] = [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ]
[ 2 2 2 2 3 3 2 4 3 2 5 3 2 6 ] [ 0 + . . . a + a q + a q + a q + (a + a ) q + (a + a ) q + (2 a + a ) q + . . . ]
[ ] [ 2 2 2 3 2 4 2 5 3 2 6 ] [ 0 + . . . a + 1 + a q + (a + a) q + (a + a) q + (2 a + a) q + (2 a + a) q + (a + 3 a + a) q + . . . ]
This came from the eensy path-invariant 3x3s {{k, {{0, q^n, 0}, {q^(k + n), q^n, 1 - q^n}, {0, 0, 1}}}, {n, {{q^(k + 2 n)/(1 - q^(1 + n)), 0, q^n}, {0, q^(k + 2 n)/(1 - q^(1 + n)), 1}, {0, 0, 1}}}}. --rwg with the usual big help from Corey&Julian
* Bill Gosper <billgosper@gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Oct 08. 2012 08:02]:> (For R-R, special-case a:=q)> > [...]> > [ inf 2 ]> [ ==== n n ]> [ \ a q ]> [ 0 a > -------------- ]> inf [ / qpoch(q, q, n) ]> /===\ [ ==== ]> | | [ 0 a ] [ n = 0 ]> (d44) | | [ ] = [ ]> | | [ k ] [ inf 2 ]> k = 0 [ q 1 ] [ ==== n n - n ]> [ \ a q ]> [ 0 > -------------- ]> [ / qpoch(q, q, n) ]> [ ==== ]> [ n = 0 ]> > [...] Jörg>I toyed around a bit (note my start k=1 of the product): [ inf 2 ] [ ==== n ] [ \ q ] [ 0 > -------------- ] (UR = H(q) = A003106) inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 1 ] [ n = 0 ] (dxx) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 1 [ q 1 ] [ ==== n + n ] [ \ q ] [ 0 > -------------- ] (LR= G(q) = A003114) [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] I.e., R-R proper. I am much tempted to put that into both A003106 and A003114 (obviously with attribution to RWG). OK, Neil ? [ inf 2 ] [ ==== n - n ] [ \ q ] [ 0 q > -------------- ] (UR=A006141) inf [ / qpoch(q, q, n) ] /===\ [ ==== ] | | [ 0 q ] [ n = 0 ] (dxx) | | [ ] = [ ] | | [ k ] [ inf 2 ] k = 1 [ q 1 ] [ ==== n ] [ \ q ] [ 0 > -------------- ] (LR= H(q) = A003106) [ / qpoch(q, q, n) ] [ ==== ] [ n = 0 ] inf /===\ | | [ 0 q^n ] [ 0 D(q) ] (D=A000009) (dxx) | | [ ] = [ ] | | [ k ] [ 0 D(q) ] (D=A000009) k = 1 [ 1+q 1 ] inf /===\ | | [ 0 -q^n ] [ 0 E(q) ] (E=A010815 = 1/A000041) (dxx) | | [ ] = [ ] | | [ k ] [ 0 E(q) ] (E=A010815 = 1/A000041) k = 1 [1-q 1 ] ASCII for president!