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%I #27 May 20 2026 04:46:53
%S 1,3,3,7,4,7,15,11,11,15,31,26,16,26,31,63,57,42,42,57,63,127,120,99,
%T 64,99,120,127,255,247,219,163,163,219,247,255,511,502,466,382,256,
%U 382,466,502,511,1023,1013,968,848,638,638,848,968,1013,1023
%N Triangle read by rows: T(n, k) = Sum_{i=0..n} binomial(2*i, i) * binomial(n - 2*i, k - i).
%C Unfortunately, there is no uniform definition of the binomial coefficient across mathematical software packages. On one side stand Maple, Mathematica, and PARI; on the other, Python, SageMath, and Sympy. The suspicion that this discrepancy has led to undetected errors cannot be dismissed, particularly in cases where code has been translated from one camp to the other. The triangle given here can serve as an example how easily a naive approach can lead to errors. For example, using Sympy's binomial function evaluates T(n, k) to A054143(n, k).
%C A definition of the binomial coefficients for Python that matches the behavior of Maple, Mathematica, and PARI is proposed in the Program section below.
%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/ExtensionsOfTheBinomial">Extensions of the Binomial</a>.
%F For a recurrence see the first Python function.
%F T(n, k) = [x^k] [t^n] (2/((-1 + 2*t)*(2*t*x - 1)) + 1/(sqrt(-4*t^2*x + 1)*(t*x + t - 1))).
%F From _Natalia L. Skirrow_, May 19 2026: (Start)
%F T(n, k) = 2^(n + 1) - A204621(n, k).
%F T(n, k) = 2^(n + 1) - binomial(n, k)*hypergeom([1/2, -k, k-n], [(1 - n)/2, -n/2], 1).
%F Let m = max(k, n-k), then:
%F T(n, k) = Sum_{i=0..m} C(n+1, i).
%F T(n, k) = Sum_{i=0..m} C(2*i, i)*C(n-2*i, k-i) (using the extension in the Prog section).
%F T(n, k) = Sum_{i=0..m} C(ceiling(n/2)-m+2*j, j)*C(floor(n/2)+m-2*j, m-j).
%F Letting m = min(k, n-k) instead produces A204621. (End)
%e Triangle starts:
%e [0] 1;
%e [1] 3, 3;
%e [2] 7, 4, 7;
%e [3] 15, 11, 11, 15;
%e [4] 31, 26, 16, 26, 31;
%e [5] 63, 57, 42, 42, 57, 63;
%e [6] 127, 120, 99, 64, 99, 120, 127;
%e [7] 255, 247, 219, 163, 163, 219, 247, 255;
%e [8] 511, 502, 466, 382, 256, 382, 466, 502, 511;
%e [9] 1023, 1013, 968, 848, 638, 638, 848, 968, 1013, 1023;
%p T := (n, k) -> local i; add(binomial(2*i, i)*binomial(n-2*i, k-i), i = 0..n):
%p seq(seq(T(n, k), k = 0..n), n = 0..9);
%p # Alternative: hypergeometric formula
%p T := (n, k) -> 2^(n + 1) - binomial(n, k)*hypergeom([1/2, -k, k-n], [(1 - n)/2, -n/2], 1): seq(print(seq(simplify(T(n, k)), k = 0..n)), n = 0..9);
%p # Alternative: g.f. of row polynomials
%p MAX := 9: G := 2/((1 - 2*t)*(1 - 2*t*x)) - 1/((1 - (1 + x)*t)*sqrt(1 - 4 * x *t^2)):
%p ser := series(G, t, MAX + 1): tcoef := n -> simplify(coeff(ser, t, n)):
%p row := n -> local k; [seq(coeff(tcoef(n), x, k), k = 0..n)]: seq(row(n), n = 0..MAX);
%t T[n_, k_] := Sum[Binomial[2 i, i] Binomial[n - 2 i, k - i], {i, 0, n}]
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}]
%o (PARI)
%o T(n, k) = sum(i = 0, n, binomial(2*i, i) * binomial(n - 2*i, k - i));
%o print_row(n) = print("n = ", n, ": ", vector(n + 1, k, T(n, k - 1)));
%o for(n = 0, 9, print_row(n))
%o (Python) # recurrence
%o def T(n: int, k: int) -> int:
%o if n == 0: return 1
%o if k == 0 or k == n: return 2*T(n-1, 0) + 1
%o if n == 2*k: return T(n-1, 0) + 1
%o return T(n-1, k-1) + T(n-1, k)
%o for n in range(10): print([n], [T(n, k) for k in range(n + 1)])
%o (Python)
%o from math import factorial as F
%o def binomial(n: int, k: int) -> int:
%o if k == 0 or k == n: return 1
%o if 0 <= k <= n: return F(n) // (F(k) * F(n - k))
%o if k <= n < 0: return F(-k - 1) // (F(n - k) * F(-1 - n)) * (-1) ** (n - k)
%o if n < 0 <= k: return F(-n - 1 + k) // (F(k) * F(-n - 1)) * (-1) ** k
%o return 0
%o def T(n, k): return sum(binomial(2 * i, i) * binomial(n - 2 * i, k - i) for i in range(n + 1))
%o for n in range(10): print([n], [T(n, k) for k in range(n + 1)])
%Y Cf. A000225 (T(n, 0) and T(n, n)), A396199 (row sums), A396198 (bisection of alternating row sums), A027306 (middle terms), A000302 (central terms).
%Y Cf. A000295, A032443, A054143, A204621.
%K nonn,tabl
%O 0,2
%A _Peter Luschny_, May 18 2026