# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a172358 Showing 1-1 of 1 %I A172358 #11 May 10 2021 03:45:57 %S A172358 1,1,1,1,1,1,1,1,1,1,1,3,3,3,1,1,3,9,9,3,1,1,5,15,45,15,5,1,1,9,45, %T A172358 135,135,45,9,1,1,11,99,495,495,495,99,11,1,1,19,209,1881,3135,3135, %U A172358 1881,209,19,1,1,29,551,6061,18183,30305,18183,6061,551,29,1 %N A172358 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of a sequence defined in comments. %C A172358 Start from the sequence A159284 and its partial products c(n) = 1, 1, 1, 1, 3, 9, 45, 405, 4455, 84645, 2454705, ... . Then T(n,k) = round( c(n)/(c(k)*c(n-k)) ). %H A172358 G. C. Greubel, Rows n = 0..50 of the triangle, flattened %F A172358 T(n, k, q) = round(c(n,q)/(c(k,q)*c(n-k,q)), where c(n,q) = Product_{j=1..n} f(j,q), f(n, q) = f(n-2, q) + q*f(n-3, q), f(0,q)=0, f(1,q) = f(2,q) = 1, and q = 2. - _G. C. Greubel_, May 09 2021 %e A172358 Triangle begins as: %e A172358 1; %e A172358 1, 1; %e A172358 1, 1, 1; %e A172358 1, 1, 1, 1; %e A172358 1, 3, 3, 3, 1; %e A172358 1, 3, 9, 9, 3, 1; %e A172358 1, 5, 15, 45, 15, 5, 1; %e A172358 1, 9, 45, 135, 135, 45, 9, 1; %e A172358 1, 11, 99, 495, 495, 495, 99, 11, 1; %e A172358 1, 19, 209, 1881, 3135, 3135, 1881, 209, 19, 1; %e A172358 1, 29, 551, 6061, 18183, 30305, 18183, 6061, 551, 29, 1; %t A172358 f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], f[n-2, q] + q*f[n-3, q]]; %t A172358 c[n_, q_]:= Product[f[j, q], {j,n}]; %t A172358 T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])]; %t A172358 Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, May 09 2021 *) %o A172358 (Sage) %o A172358 @CachedFunction %o A172358 def f(n,q): return fibonacci(n) if (n<3) else f(n-2, q) + q*f(n-3, q) %o A172358 def c(n,q): return product( f(j,q) for j in (1..n) ) %o A172358 def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q))) %o A172358 flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 09 2021 %Y A172358 Cf. A172353 (q=1), this sequence (q=2), A172359 (q=4), A172360 (q=5). %K A172358 nonn,tabl,less %O A172358 0,12 %A A172358 _Roger L. Bagula_, Feb 01 2010 %E A172358 Definition corrected to give integral terms by _G. C. Greubel_, May 09 2021 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE