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%I #13 Jun 04 2026 05:36:37
%S 0,1,3,15,88,563,3812,26877,195349,1453763,11024089,84885641,
%T 661948987,5217188125,41493928665,332601055544,2684193028375,
%U 21791893755523,177856290438261,1458434632623471,12009839989925178,99275036681040765,823456735417367135
%N 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.
%C 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.
%H Nathan Fox, <a href="/A394151/b394151.txt">Table of n, a(n) for n = 0..300</a>
%H S. Dimitrov, N. Fox, K. Hadaway, A. Tharp, and S. Wagner, <a href="https://arxiv.org/abs/2602.16055">Counting Colored Trees</a>, arXiv:2602.16055 [math.CO], 2026.
%F 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
%F 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
%p 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):
%p map(f, [$0..30]); # _Robert Israel_, Mar 12 2026
%o (Python)
%o def A394151(n):
%o A = [[1, 1, 1], [1, 0, 1], [0, 0, 1]]
%o if n == 0:
%o return 0
%o m = len(A)
%o output = [[1] for i in range(m)]
%o for l in range(2, n + 1):
%o for i in range(m):
%o term = 0
%o for k in range(1, l):
%o for j in range(m):
%o term += A[i][j] * output[i][k - 1] * output[j][l - k - 1]
%o output[i].append(term)
%o return output[0][n - 1]
%K nonn
%O 0,3
%A _Nathan Fox_, Mar 12 2026