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A374441 Triangle read by rows: T(n, k) = binomial(n - floor(k/2), ceiling(k/2)) - binomial(n - ceiling(k/2), floor(k/2)).

%I A374441 #41 Jan 05 2025 19:51:42
%S A374441 0,0,0,0,1,0,0,2,0,0,0,3,0,1,0,0,4,0,3,0,0,0,5,0,6,0,1,0,0,6,0,10,0,4,
%T A374441 0,0,0,7,0,15,0,10,0,1,0,0,8,0,21,0,20,0,5,0,0,0,9,0,28,0,35,0,15,0,1,
%U A374441 0,0,10,0,36,0,56,0,35,0,6,0,0,0,11,0,45,0,84,0,70,0,21,0,1,0
%N A374441 Triangle read by rows: T(n, k) = binomial(n - floor(k/2), ceiling(k/2)) - binomial(n - ceiling(k/2), floor(k/2)).
%C A374441 Member of the family of Fibonacci polynomials (A011973, A162515, ...) and Chebyshev polynomials (A053119).
%H A374441 Paolo Xausa, <a href="/A374441/b374441.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of triangle, flattened).
%H A374441 Henry W. Gould, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/3-4/gould.pdf">A Variant of Pascal's Triangle</a>, The Fibonacci Quarterly, Vol. 3, Nr. 4, Dec. 1965, pp. 257-271, with <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/4-1/corrections2.pdf">corrections</a>.
%F A374441 T(n, k) = [x^(n-k)][z^n] (x / (1 - x*z - z^2)).
%F A374441 T(n, k) = binomial(n - (k + 1)/2, (k + 1)/2) if k is odd, and otherwise 0.
%F A374441 Sum_{k=0..n} T(n, k) = Fibonacci(n + 1) - 1.
%F A374441 Columns with odd index agree with the odd indexed columns of A374440.
%e A374441 Triangle starts:
%e A374441   [ 0]  0;
%e A374441   [ 1]  0, 0;
%e A374441   [ 2]  0, 1, 0;
%e A374441   [ 3]  0, 2, 0,  0;
%e A374441   [ 4]  0, 3, 0,  1, 0;
%e A374441   [ 5]  0, 4, 0,  3, 0,  0;
%e A374441   [ 6]  0, 5, 0,  6, 0,  1, 0;
%e A374441   [ 7]  0, 6, 0, 10, 0,  4, 0,  0;
%e A374441   [ 8]  0, 7, 0, 15, 0, 10, 0,  1, 0;
%e A374441   [ 9]  0, 8, 0, 21, 0, 20, 0,  5, 0, 0;
%e A374441   [10]  0, 9, 0, 28, 0, 35, 0, 15, 0, 1, 0;
%p A374441 T := (n, k) -> if k::even then 0 else binomial(n - (k + 1)/2, (k + 1)/2) fi:
%p A374441 # Or as a recurrence:
%p A374441 T := proc(n, k) option remember; if k::even or k > n then 0 elif k = 1 then n - 1 else T(n - 1, k) + T(n - 2, k - 2) fi end:
%p A374441 seq(seq(T(n, k), k = 0..n), n = 0..12);
%t A374441 A374441[n_, k_] := If[OddQ[k], Binomial[n - (k + 1)/2, (k + 1)/2], 0];
%t A374441 Table[A374441[n, k], {n, 0, 15}, {k, 0, n}] (* _Paolo Xausa_, Nov 16 2024 *)
%o A374441 (Python)
%o A374441 from math import isqrt, comb
%o A374441 def A374441(n):
%o A374441     a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
%o A374441     b = n-comb(a+1,2)
%o A374441     return comb(a-(b+1>>1),b+1>>1) if b&1 else 0 # _Chai Wah Wu_, Nov 14 2024
%o A374441 (Python)
%o A374441 from math import comb as binomial
%o A374441 def row(n: int) -> list[int]:
%o A374441     return [binomial(n - (k+1)//2, (k+1)//2) if k%2 else 0 for k in range(n+1)]
%o A374441 for n in range(11): print(row(n))  # _Peter Luschny_, Nov 21 2024
%Y A374441 Cf. A374440 (odd columns agree).
%Y A374441 Cf. A000071 (row sums), A065941, A194005, A103631, A007318.
%Y A374441 Cf. A011973, A162515, A053119.
%K A374441 nonn,tabl
%O A374441 0,8
%A A374441 _Peter Luschny_, Jul 19 2024