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

2,2

Links

Table of n, a(n) for n=2..29.

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).

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

Example

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.

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]

Crossrefs

Cf. A375853.

Sequence in context: A246134 A115112 A171074 * A005367 A050184 A263582

Adjacent sequences: A376069 A376070 A376071 * A376073 A376074 A376075

Keyword

nonn,easy

Author

Mingjian Ding, Sep 08 2024

Status

approved