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Andrew Howroyd, <a href="/A335362/b335362.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows).

1, 1, 1, 1, 2, 3, 2, 5, 10, 8, 3, 12, 32, 40, 27, 6, 30, 99, 178, 187, 91, 11, 74, 298, 692, 1019, 854, 350, 23, 188, 890, 2538, 4751, 5692, 4074, 1376, 47, 478, 2627, 8886, 20260, 31188, 31856, 19602, 5743, 106, 1235, 7734, 30270, 81170, 152509, 200413, 177266, 96035, 24635

Andrew Howroyd, <a href="/A335362/b335362.txt">Table of n, a(n) for n = 1..1275</a>

(PARI) \\ Here R(n) is rooted mixed trees as g.f.

EulerMTS(p)={my(n=serprec(p, x)-1, vars=variables(p)); exp(sum(i=1, n, substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i))}

R(n) = {my(p=x+O(x^2)); for(n=2, n, p=x*EulerMTS(2*y*p + p)); p}

T(n) = {my(p=R(n)); [Vecrev(p) | p<-Vec(p + (subst(subst(p + O(x*x^(n\2)), x, x^2), y, y^2) - (2*y+1)*p^2)/2)]}

{ my(A=T(10)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Mar 23 2023

Terms a(46) and beyond from Andrew Howroyd, Mar 23 2023

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Cf. A000055 (column d=0), A000238 (diagonal d=n-1), A000106 (column d=1), A006965 (row sums), A335601 (diagonal subdiagonal d=n-2).

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The generating function of the first subdiagonal A(x) = x^2 +2x^3 +10x^4 +40x^5 +187x^6 +... = (B(x)^2+B(x^2))/2, where B(x) is the g.f. of A000151. - R. J. Mathar, Jun 04 2020

Cf. A000055 (column d=0), A000238 (diagonal n=d+=n-1), A000106 (column d=1), A006965 (row sums), A335601 (diagonal d=n-2).

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1, 1, 1, 1, 2, 3, 2, 5, 10, 8, 3, 12, 32, 40, 27, 6, 30, 99, 178, 187, 91, 11, 74, 298, 692, 1019, 854, 350, 23, 188, 890, 2538, 4751, 5692, 4074, 1376, 47, 478, 2627, 8886, 20260, 31188, 31856, 19602, 5743

Completed row n=9. - R. J. Mathar, Jun 11 2020

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