A054450 - OEIS

%I #18 Jul 26 2022 10:09:50

%S 1,1,1,2,1,1,3,3,1,1,5,4,4,1,1,8,8,5,5,1,1,13,12,12,6,6,1,1,21,21,17,

%T 17,7,7,1,1,34,33,33,23,23,8,8,1,1,55,55,50,50,30,30,9,9,1,1,89,88,88,

%U 73,73,38,38,10,10,1,1,144,144,138,138,103,103,47,47,11,11,1,1

%N Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials).

%C In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is Fib(z)/(1-x*z/(1-z^2)) where Fib(x)=1/(1-x-x^2) = g.f. for A000045(n+1) (Fibonacci numbers without 0).

%C This is the first member of the family of Riordan-type matrices obtained from the unsigned convolution matrix A049310 by repeated application of the partial row sums procedure.

%H G. C. Greubel, <a href="/A054450/b054450.txt">Rows n = 0..50 of the triangle, flattened</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%F T(n, m) = Sum_{k=m..n} |A049310(n, k)| (sequence of partial row sums in column m).

%F Column m recursion: T(n, m) = Sum_{j=m..n} T(j-1, m)*|A049310(n-j, 0)| + |A049310(n, m)|, n >= m >= 0, a(n, m) := 0 if n<m.

%F T(n, 0) = A000045(n+1).

%F T(n, 1) = A052952(n-1).

%F T(n, 2) = A054451(n-2).

%F Sum_{k=0..n} T(n, k) = A029907(n) = A054453(n, 0).

%F G.f. for column m: Fib(x)*(x/(1-x^2))^m, m >= 0, with Fib(x) = g.f. A000045(n+1).

%F The corresponding square array has T(n, k) = Sum_{j=0..floor(k/2)} binomial(n+k-j, j). - _Paul Barry_, Oct 23 2004

%F From _G. C. Greubel_, Jul 25 2022: (Start)

%F T(n, 3) = A099571(n-3).

%F T(n, 4) = A099572(n-4).

%F T(n, n) = T(n, n-1) = A000012(n).

%F T(n, n-2) = A000027(n), n >= 2.

%F T(n, n-3) = A000027(n), n >= 3.

%F T(n, n-4) = A152948(n), n >= 4.

%F T(n, n-5) = A152948(n), n >= 5.

%F T(n, n-6) = A038793(n), n >= 6.

%F T(n, n-8) = A038794(n), n >= 8.

%F T(n, n-10) = A038795(n), n >= 10.

%F T(n, n-12) = A038796(n), n >= 12. (End)

%e Triangle begins as:

%e    1;

%e    1,  1;

%e    2,  1,  1;

%e    3,  3,  1,  1;

%e    5,  4,  4,  1,  1;

%e    8,  8,  5,  5,  1,  1;

%e   13, 12, 12,  6,  6,  1,  1;

%e   21, 21, 17, 17,  7,  7,  1,  1;

%e   34, 33, 33, 23, 23,  8,  8,  1,  1;

%e   55, 55, 50, 50, 30, 30,  9,  9,  1, 1;

%e   89, 88, 88, 73, 73, 38, 38, 10, 10, 1, 1;

%e   ...

%e Fourth row polynomial (n=3): p(3,x) = 3 + 3*x + x^2 + x^3.

%t A049310[n_, k_]:= A049310[n, k]= If[n<0, 0, If[k==n, 1, A049310[n-1, k-1] - A049310[n-2, k] ]];

%t A054450[n_, k_]:= A054450[n, k]= Sum[Abs[A049310[n,j]], {j,k,n}];

%t Table[A054450[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jul 25 2022 *)

%o (Magma)

%o A049310:= func< n,k | ((n+k) mod 2) eq 0 select (-1)^(Floor((n+k)/2)+k)*Binomial(Floor((n+k)/2), k) else 0 >;

%o A054450:= func< n,k | (&+[Abs(A049310(n,j)): j in [k..n]]) >;

%o [A054450(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Jul 25 2022

%o (SageMath)

%o @CachedFunction

%o def A049310(n, k):

%o     if (n<0): return 0

%o     elif (k==n): return 1

%o     else: return A049310(n-1, k-1) - A049310(n-2, k)

%o def A054450(n,k): return sum( abs(A049310(n,j)) for j in (k..n) )

%o flatten([[A054450(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Jul 25 2022

%Y Cf. A000027, A000045, A029907 (row sums), A038793, A038794, A038795, A038796.

%Y Cf. A049310, A052952, A053121, A054451, A054453, A099571, A099572, A152948.

%K easy,nonn,tabl

%O 0,4

%A _Wolfdieter Lang_, Apr 27 2000 and May 08 2000