A379103 Expansion of (1-3*x-sqrt(9*x^2-14*x+1))/4.

0, 1, 5, 35, 295, 2765, 27705, 290535, 3148995, 34995065, 396602605, 4566227435, 53259218495, 627982592965, 7473163652705, 89640387354735, 1082664905352795, 13155505626756465, 160709002086562005, 1972595405313408435, 24315686632846439895, 300886761671728853565, 3736205372071338170505, 46540791299676591116535

Offset

0,3

Comments

Problem A6 on the 2024 William Lowell Putnam Mathematical Competition was to compute the Hankel transform of this sequence, which is A110147.

Links

Table of n, a(n) for n=0..23.

Nathaniel Johnston, Determinant involving (1 - 3x - sqrt(1 - 14x + 9x^2))/4, YouTube video, 2024.

Nathaniel Johnston, Putnam 2024 A6 solution, 2024.

Formula

a(0) = 0, a(1) = 1, a(n) = 3*a(n-1) + 2*Sum_{k=0..n} a(k)*a(n-k) for n >= 2.

G.f.: (1-3*x-sqrt(9*x^2-14*x+1))/4.

G.f.: x/(1-5*x/(1-2*x/(1-5*x/(1-2*x/(1-5*x/(...)))))). - Thomas Scheuerle, Feb 28 2025

a(n) = (1/4)*(-1)^(n+1) * Sum_{k=0..n} binomial(1/2,k) * binomial(1/2,n-k) * (7+2*sqrt(10))^k * (7-2*sqrt(10))^(n-k) for n >= 2. - Ehren Metcalfe, Feb 26 2025

a(n) ~ 5^(1/4) * (7 + 2*sqrt(10))^(n - 1/2) / (2^(7/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 27 2025

Prog

(MATLAB) a = 3; b = 2; c(1) = 1; last_val = 16; for j = 2:last_val
c(j) = a*c(j-1) + b*sum(c(1:j-1).*fliplr(c(1:j-1)));
end
(PARI) my(x='x+O('x^33)); concat([0], Vec((1-3*x-sqrt(9*x^2-14*x+1))/4)) \\ Joerg Arndt, Dec 15 2024

Crossrefs

Cf. A025231, A047891, A082298, A110147.

Sequence in context: A051406 A000356 A370844 * A027392 A291813 A346765

Adjacent sequences: A379100 A379101 A379102 * A379104 A379105 A379106

Keyword

nonn,easy,changed

Author

Nathaniel Johnston, Dec 15 2024

Status

approved