A376072 a(n) are half the sums of the gamma coefficients of the n-th row-generating function of triangle A375853.

1, 4, 18, 68, 251, 888, 3076, 10456, 35061, 116252, 381974, 1245564, 4035631, 13003696, 41701512, 133175792, 423741161, 1343864820, 4249518490, 13402327540, 42168298851, 132388845224, 414818381708, 1297410683208, 4051098663901, 12629895834508, 39319487031966, 122247859681196

Offset: 2

Author

Mingjian Ding, Sep 08 2024

Keywords

• nonn
• easy

Examples

			For n = 4, the row-generating function of triangle A375853(n, k) is 20*x + 56*x^2 + 20*x^3. Thus the corresponding gamma polynomial is 20*x + 16*x^2, and so a(4) = 18.
		

Links

• Ming-Jian Ding and Jiang Zeng, Some new results on minuscule polynomial of type A, arXiv:2308.16782, [math.CO], 2023.
• Index entries for linear recurrences with constant coefficients, signature (4,2,-12,-9).

Crossrefs

Cf. A375853.

Programs

Maple

a := n -> (3^n*(2*n - 1) + (-1)^n*(2*n + 1))/32:
seq(a(n), n = 2..19);  # Peter Luschny, Sep 23 2024

Mathematica

LinearRecurrence[{4,2,-12,-9},{1,4,18,68},30]

Formula

a(n) = 2^(n-1)*T_n((-1 + sqrt(3)*i)/2)/(1 + sqrt(3)*i)^n, where T_n(x) is the generating function of the n-th row of A375853.

a(n) = a(n - 1) + n*(3^(n-1) + (-1)^n)/8, a(2) = 1.

a(n) = ((2*n - 2)*a(n - 1) + 3*n*a(n - 2))/(n - 2), a(2) = 1, a(3) = 4.

a(n) = ((2*n - 1)*3^n + (2*n + 1)*(-1)^n)/32.

G.f.: x^2/(1 - 2*x - 3*x^2)^2.

E.g.f.: exp(x)*(2*x*cosh(2*x) - (1 - 4*x)*sinh(2*x))/16. - Stefano Spezia, Sep 23 2024