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A155558 Multi-bifurcating recursion of a factorial type based on the MacMahon numbers A060187 as a triangle sequence: t(n,k) = A060187(n,m) from polynomials; f(n, m) = If[m <= Floor[n/2], f(m, 1)*f(n - m, 1)*t(n + 1, m)].

%I A155558 #7 Jun 01 2025 18:28:48
%S A155558 1,1,6,138,10488,8280,2485656,1392696,1794643632,663449904,448448112,
%T A155558 3910528474128,904634615952,375908525712,25621782562486656,
%U A155558 3570752184568704,852504801992064,514242863174016,504057328351799983488
%N A155558 Multi-bifurcating recursion of a factorial type based on the MacMahon numbers A060187 as a triangle sequence: t(n,k) = A060187(n,m) from polynomials; f(n, m) = If[m <= Floor[n/2], f(m, 1)*f(n - m, 1)*t(n + 1, m)].
%C A155558 Row sums are: {1, 1, 6, 138, 18768, 3878352, 2906541648, 5191071615792, 30559282412221440, 552452422047369288192, 31282693282240728431406336,...}.
%C A155558 The Eulerian numbers factored as factorial like to middle Floor[n/2]: t(n,m)=f(n,m)/(f[m,1]*f[n-m,1]).
%C A155558 The idea is to factor the Eulerian numbers as if the coefficients were made up of equivalents to factorials.
%C A155558 The result is a multi-bifurcating recursion thast fits the Eulerian numbers.
%F A155558 t(n,k)=A060187[n,m] from polynomials; f(n, m) = If[m <= Floor[n/2], f(m, 1)*f(n - m, 1)*t(n + 1, m)];
%e A155558 Half MacMahon numbers: Table[Table[f[n, m]/(f[m, 1]*f[n - m, 1]), {m, 0, Floor[n/2]}], {n, 0, 10}];
%e A155558   {1},
%e A155558   {1},
%e A155558   {1, 6},
%e A155558   {1, 23},
%e A155558   {1, 76, 230},
%e A155558   {1, 237, 1682},
%e A155558   {1, 722, 10543, 23548},
%e A155558   {1, 2179, 60657, 259723},
%e A155558   {1, 6552, 331612, 2485288, 4675014},
%e A155558   {1, 19673, 1756340, 21707972, 69413294},
%e A155558   {1, 59038, 9116141, 178300904, 906923282, 1527092468}
%e A155558 Factorial type triangle is:
%e A155558   {1},
%e A155558   {1},
%e A155558   {6},
%e A155558   {138},
%e A155558   {10488, 8280},
%e A155558   {2485656, 1392696},
%e A155558   {1794643632, 663449904, 448448112},
%e A155558   {3910528474128, 904634615952, 375908525712},
%e A155558   {25621782562486656, 3570752184568704, 852504801992064, 514242863174016},
%e A155558   ...
%t A155558 Clear[t, n, m, f, x, p];
%t A155558 p[x_, n_] = (-1)^(n + 1)*(x - 1)^(n + 1)*Sum[(2* m + 1)^n*x^m, {m, 0, Infinity}];
%t A155558 t[n_, m_] := Table[CoefficientList[FullSimplify[ExpandAll[p[x, k]]], x], { k, 0, 10}][[n + 1, m + 1]];
%t A155558 f[0, 1] = 1; f[1, 1] = 1; f[2, 1] = 6;
%t A155558 f[n_, m_] := f[n, m] = If[m <= Floor[n/2], f[m, 1]*f[n - m, 1]*t[n, m]];
%t A155558 a = Join[{{1}}, {{1}}, Table[Table[f[n, m], {m, 1, Floor[n/2]}], {n, 2, 10}]];
%t A155558 Flatten[%]
%Y A155558 Cf. A008292, A060187.
%K A155558 nonn,tabf,uned
%O A155558 0,3
%A A155558 _Roger L. Bagula_, Jan 24 2009