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A376072
a(n) are half the sums of the gamma coefficients of the n-th row-generating function of triangle A375853.
1
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
Ming-Jian Ding and Jiang Zeng, Some new results on minuscule polynomial of type A, arXiv:2308.16782, [math.CO], 2023.
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
KEYWORD
nonn,easy
AUTHOR
Mingjian Ding, Sep 08 2024
STATUS
approved