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A055830 Triangle T read by rows: diagonal differences of triangle A037027.

%I A055830 #35 Jan 05 2025 19:51:36
%S A055830 1,1,0,2,1,0,3,3,1,0,5,7,4,1,0,8,15,12,5,1,0,13,30,31,18,6,1,0,21,58,
%T A055830 73,54,25,7,1,0,34,109,162,145,85,33,8,1,0,55,201,344,361,255,125,42,
%U A055830 9,1,0,89,365,707,850,701,413,175,52,10,1,0,144,655,1416,1918,1806,1239,630,236,63,11,1,0
%N A055830 Triangle T read by rows: diagonal differences of triangle A037027.
%C A055830 Or, coefficients of a generalized Lucas-Pell polynomial read by rows. - _Philippe Deléham_, Nov 05 2006
%C A055830 Equals A046854(shifted) * Pascal's triangle; where A046854 is shifted down one row and "1" inserted at (0,0). - _Gary W. Adamson_, Dec 24 2008
%H A055830 G. C. Greubel, <a href="/A055830/b055830.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%H A055830 Clark Kimberling, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/40-4/kimberling.pdf">Path-counting and Fibonacci numbers</a>, Fib. Quart. 40 (4) (2002) 328-338, Example 3D.
%F A055830 G.f.: (1-y*z) / (1-y*(1+y+z)).
%F A055830 T(i, j) = R(i-j, j), where R(0, 0)=1, R(0, j)=0 for j >= 1, R(1, j)=1 for j >= 0, R(i, j) = Sum_{k=0..j} (R(i-2, k) + R(i-1, k)) for i >= 1, j >= 1.
%F A055830 Sum_{k=0..n} x^k*T(n,k) = A039834(n-2), A000012(n), A000045(n+1), A001333(n), A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), A015455(n), A015456(n), A015457(n) for x= -2,-1,0,1,2,3,4,5,6,7,8,9,10. - _Philippe Deléham_, Oct 22 2006
%F A055830 Sum_{k=0..floor(n/2)} T(n-k,k) = A011782(n). - _Philippe Deléham_, Oct 22 2006
%F A055830 Triangle T(n,k), 0 <= k <= n, given by [1, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Nov 05 2006
%F A055830 T(n,0) = Fibonacci(n+1) = A000045(n+1). Sum_{k=0..n} T(n,k) = A001333(n). T(n,k)=0 if k > n or if k < 0, T(0,0)=1, T(1,1)=0, T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-2,k). - _Philippe Deléham_, Nov 05 2006
%e A055830 Triangle begins:
%e A055830    1
%e A055830    1,   0
%e A055830    2,   1,   0
%e A055830    3,   3,   1,   0
%e A055830    5,   7,   4,   1,   0
%e A055830    8,  15,  12,   5,   1,   0
%e A055830   13,  30,  31,  18,   6,   1,  0
%e A055830   21,  58,  73,  54,  25,   7,  1, 0
%e A055830   34, 109, 162, 145,  85,  33,  8, 1, 0
%e A055830   55, 201, 344, 361, 255, 125, 42, 9, 1, 0
%e A055830   ...
%p A055830 with(combinat);
%p A055830 T:= proc(n, k) option remember;
%p A055830       if k<0 or k>n then 0
%p A055830     elif k=0 then fibonacci(n+1)
%p A055830     elif n=1 and k=1 then 0
%p A055830     else T(n-1, k-1) + T(n-1, k) + T(n-2, k)
%p A055830       fi; end:
%p A055830 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Jan 21 2020
%t A055830 T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[k==0, Fibonacci[n+1], If[n==1 && k==1, 0, T[n-1, k-1] + T[n-1, k] + T[n-2, k]]]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Dec 19 2017 *)
%o A055830 (PARI) T(n,k) = if(k<0 || k>n, 0, if(k==0, fibonacci(n+1), if(n==1 && k==1, 0, T(n-1, k-1) + T(n-1, k) + T(n-2, k) )));
%o A055830 for(n=0,12, for(k=0, n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jan 21 2020
%o A055830 (Magma)
%o A055830 function T(n,k)
%o A055830   if k lt 0 or k gt n then return 0;
%o A055830   elif k eq 0 then return Fibonacci(n+1);
%o A055830   elif n eq 1 and k eq 1 then return 0;
%o A055830   else return T(n-1,k-1) + T(n-1,k) + T(n-2,k);
%o A055830   end if; return T; end function;
%o A055830 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 21 2020
%o A055830 (Sage)
%o A055830 @CachedFunction
%o A055830 def T(n, k):
%o A055830     if (k<0 or k>n): return 0
%o A055830     elif (k==0): return fibonacci(n+1)
%o A055830     elif (n==1 and k==1): return 0
%o A055830     else: return T(n-1, k-1) + T(n-1, k) + T(n-2, k)
%o A055830 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jan 21 2020
%Y A055830 Left-hand columns include A000045, A023610.
%Y A055830 Right-hand columns include A055831, A055832, A055833, A055834, A055835, A055836, A055837, A055838, A055839, A055840.
%Y A055830 Row sums: A001333 (numerators of continued fraction convergents to sqrt(2)).
%Y A055830 Cf. A122075 (another version).
%Y A055830 Cf. A046854. - _Gary W. Adamson_, Dec 24 2008
%K A055830 nonn,tabl
%O A055830 0,4
%A A055830 _Clark Kimberling_, May 28 2000
%E A055830 Edited by _Ralf Stephan_, Jan 12 2005