%I #31 Mar 31 2023 17:33:49

%S 1,1,2,2,9,9,6,44,96,64,24,250,875,1250,625,120,1644,8100,18360,19440,

%T 7776,720,12348,79576,252105,420175,352947,117649,5040,104544,840448,

%U 3465728,8028160,10551296,7340032,2097152

%N Triangle of numbers related to rooted trees and unrooted planar trees.

%C The rows sum to A006963: (2*n - 1)!/n!.

%C The main diagonal is A000169: n^(n-1).

%C The left column is A000142: (n - 1)!.

%C The alternating sum in row n is (-1)^(n-1)*(n - 1)!

%C If Y := X * (1 - X)^(z-1), then (1 - z*X)^(-1) = 1 + Sum_{n>=1} Y^n/(n-1)! * (Sum_{k=1..n} (-1)^(n-k) * z^k * T(n, k)). Note that if Y = y^(z-1) and X = x^(z-1) then y = x - x^z, dy/dx = 1 - z*x^(z-1) = 1 - z*X, and dx/dy = (1 - z*X)^(-1). Also x = y + x^z = y + y^z + z*y^(2*z-1) + ... = y * (1 + Sum_{n>=1} Y^n/(n-1)! * (1+(z-1)*n)^(-1) * (Sum_{k=1..n} (-1)^(n-k) * z^k * T(n, k))). - _Michael Somos_, Aug 01 2019

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1998.

%F Formula for row n: Sum_{k = 0..n-1} T(n,k)*y^k = Product_{k = 1..n-1} (k + n*y)

%F E.g.f.: A(x,t) = Sum_{n >= 1} 1/(n*t)*binomial(n*t + n - 1, n)*x^n = log(B_(t+1)(x)), where B_t(x) = Sum_{n >= 0} 1/(n*t + 1)*binomial(n*t + 1, n)*x^n is Lambert's generalized binomial series - see Graham et al., Section 5.4. - _Peter Bala_, Nov 08 2015

%F T(n,m) = n^(m-1)*binomial(n-1,m-1)*Sum_{k=0..n-m} ((-1)^(n-m-k)*binomial(n+k-1,k)*stirling2(n-m+k,k)*binomial(2*n-m,n-m-k))/binomial(n-m+k,k). - _Vladimir Kruchinin_, Apr 05 2016

%F Conjecture: T(n,k) = A130534(n,k)* n^(k-1). - _R. J. Mathar_, Mar 31 2023

%e Triangle begins:

%e {1},

%e {1, 2},

%e {2, 9, 9},

%e {6, 44, 96, 64},

%e {24, 250, 875, 1250, 625},

%e ...

%p seq(seq(coeff(product(n*x + k, k = 1..n-1), x, i), i = 0..n-1), n = 1..8); # _Peter Bala_, Nov 08 2015

%t T[n_, m_] := (n^(m-1)*Binomial[n-1, m-1]*Sum[((-1)^(n-m-k)*Binomial[n+k-1, k]*StirlingS2[n-m+k, k]*Binomial[2*n-m, n-m-k])/Binomial[n-m+k, k], {k, 0, n-m}]); Table[T[n, m], {n, 1, 8}, {m, 1, n}] // Flatten (* Jean-François Alcover, Feb 23 2017, after Vladimir Kruchinin *)

%t T[ n_, k_] := If[ n < 1 || k < 1, 0, Coefficient[ (-1)^(n - k) Binomial[n z, n] (n - 1)!, z, k]]; (* _Michael Somos_, Aug 01 2019 *)

%o (Maxima)

%o T(n,m):=(n^(m-1)*binomial(n-1,m-1)*sum(((-1)^(n-m-k)*binomial(n+k-1,k)*stirling2(n-m+k,k)*binomial(2*n-m,n-m-k))/binomial(n-m+k,k),k,0,n-m)); /* _Vladimir Kruchinin_, Apr 05 2016 */

%o (PARI) {T(n, k) = if( n < 1 || k < 1, 0, polcoeff( (-1)^(n-k) * binomial(n*x, n)*(n-1)!, k))}; /* _Michael Somos_, Aug 01 2019 */

%Y Cf. A006963, A000142, A000169, A203904.

%K nonn,tabl,easy

%O 1,3

%A _F. Chapoton_, Aug 31 2000

%E a(29)-a(36) from _Peter Bala_, Nov 08 2015