A054450 Triangle of partial row sums of unsigned triangle A049310(n,m), n >= m >= 0 (Chebyshev S-polynomials).

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

Offset

0,4

Comments

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).

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.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Index entries for sequences related to Chebyshev polynomials.

Formula

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

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.

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

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

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

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

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

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

From G. C. Greubel, Jul 25 2022: (Start)

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

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

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

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

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

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

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

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

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

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

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

Example

Triangle begins as:
   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, 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, 73, 73, 38, 38, 10, 10, 1, 1;
  ...
Fourth row polynomial (n=3): p(3,x) = 3 + 3*x + x^2 + x^3.

Mathematica

A049310[n_, k_]:= A049310[n, k]= If[n<0, 0, If[k==n, 1, A049310[n-1, k-1] - A049310[n-2, k] ]];
A054450[n_, k_]:= A054450[n, k]= Sum[Abs[A049310[n, j]], {j, k, n}];
Table[A054450[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 25 2022 *)

Prog

(Magma)
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 >;
A054450:= func< n, k | (&+[Abs(A049310(n, j)): j in [k..n]]) >;
[A054450(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jul 25 2022
(SageMath)
@CachedFunction
def A049310(n, k):
    if (n<0): return 0
    elif (k==n): return 1
    else: return A049310(n-1, k-1) - A049310(n-2, k)
def A054450(n, k): return sum( abs(A049310(n, j)) for j in (k..n) )
flatten([[A054450(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Jul 25 2022

Crossrefs

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

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

Sequence in context: A137896 A157219 A167040 * A344610 A337009 A174802

Adjacent sequences: A054447 A054448 A054449 * A054451 A054452 A054453

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang, Apr 27 2000 and May 08 2000

Status

approved