A394151 G.f. A(x) satisfies A(x)^6-2*A(x)^5+(2x+1)*A(x)^4-3x*A(x)^3+5x^2*A(x)^2-x^2*A(x)+x^3=0.

0, 1, 3, 15, 88, 563, 3812, 26877, 195349, 1453763, 11024089, 84885641, 661948987, 5217188125, 41493928665, 332601055544, 2684193028375, 21791893755523, 177856290438261, 1458434632623471, 12009839989925178, 99275036681040765, 823456735417367135

Offset

0,3

Comments

Number of n-vertex planar rooted trees with vertices colored red, blue, and green with red root where red vertices can be followed by vertices of any colors, blue vertices can be followed by red or green vertices, and green vertices can only be followed by green vertices.

Links

Nathan Fox, Table of n, a(n) for n = 0..300

S. Dimitrov, N. Fox, K. Hadaway, A. Tharp, and S. Wagner, Counting Colored Trees, arXiv:2602.16055 [math.CO], 2026.

Formula

D-finite with recurrence: -125122200*n*(2*n + 3)*(2*n + 1)*(2*n - 1)*(n + 1)*a(n) + 372*(2*n + 3)*(2*n + 1)*(n + 1)*(527797*n^2 + 1301371*n + 902790)*a(n + 1) - 2*(2*n + 3)*(51164561*n^4 + 329231564*n^3 + 755581003*n^2 + 691604500*n + 175307700)*a(n + 2) + 3*(5119353*n^5 + 38894737*n^4 + 51079029*n^3 - 282837985*n^2 - 942818190*n - 810435600)*a(n + 3) + 6*(194421*n^5 + 4486310*n^4 + 38490857*n^3 + 157435038*n^2 + 311112030*n + 239491800)*a(n + 4) - 4*(n + 5)*(36439*n^4 + 572369*n^3 + 3354049*n^2 + 8690169*n + 8401140)*a(n + 5) + 24*(2*n + 11)*(3*n + 16)*(n + 6)*(n + 5)*(3*n + 14)*a(n + 6) = 0. - Robert Israel, Mar 12 2026

a(n) ~ 31^(2*n - 1/2) / (sqrt(23*Pi) * n^(3/2) * 2^(2*n) * 3^(3*n - 3/2)). - Vaclav Kotesovec, Jun 04 2026

Maple

f:= gfun:-rectoproc({(-1000977600*n^5 - 2502444000*n^4 - 1251222000*n^3 + 625611000*n^2 + 375366600*n)*a(n) + (785361936*n^5 + 4292525856*n^4 + 9312416988*n^3 + 9944286144*n^2 + 5146546716*n + 1007513640)*a(n + 1) + (-204658244*n^5 - 1623913622*n^4 - 4997713396*n^3 - 7299904018*n^2 - 4850857800*n - 1051846200)*a(n + 2) + (15358059*n^5 + 116684211*n^4 + 153237087*n^3 - 848513955*n^2 - 2828454570*n - 2431306800)*a(n + 3) + (1166526*n^5 + 26917860*n^4 + 230945142*n^3 + 944610228*n^2 + 1866672180*n + 1436950800)*a(n + 4) + (-145756*n^5 - 3018256*n^4 - 24863576*n^3 - 101841656*n^2 - 207407940*n - 168022800)*a(n + 5) + (432*n^5 + 11448*n^4 + 121128*n^3 + 639648*n^2 + 1685856*n + 1774080)*a(n + 6), a(0) = 0, a(1) = 1, a(2) = 3, a(3) = 15, a(4) = 88, a(5) = 563}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 12 2026

Prog

(Python)
def A394151(n):
    A = [[1, 1, 1], [1, 0, 1], [0, 0, 1]]
    if n == 0:
        return 0
    m = len(A)
    output = [[1] for i in range(m)]
    for l in range(2, n + 1):
        for i in range(m):
            term = 0
            for k in range(1, l):
                for j in range(m):
                    term += A[i][j] * output[i][k - 1] * output[j][l - k - 1]
            output[i].append(term)
    return output[0][n - 1]

Crossrefs

Sequence in context: A316666 A192253 A394122 * A368972 A370472 A365128

Adjacent sequences: A394148 A394149 A394150 * A394152 A394153 A394154

Keyword

nonn

Author

Nathan Fox, Mar 12 2026

Status

approved