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A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3*k} (n! * binomial(n - 3*k, j)) / (k!*(n - 3*k)!*3^k).

%I A344912 #9 Jun 07 2021 04:33:59
%S A344912 1,1,1,1,2,1,1,3,3,1,2,1,4,6,4,1,8,8,1,5,10,10,5,1,20,40,20,1,6,15,20,
%T A344912 15,6,1,40,120,120,40,40,1,7,21,35,35,21,7,1,70,280,420,280,70,280,
%U A344912 280,1,8,28,56,70,56,28,8,1,112,560,1120,1120,560,112,1120,2240,1120
%N A344912 Irregular triangle read by rows, Trow(n) = Seq_{k=0..n/3} Seq_{j=0..n-3*k} (n! * binomial(n - 3*k, j)) / (k!*(n - 3*k)!*3^k).
%C A344912 Consider a sequence of Pascal tetrahedrons (depending on a parameter m >= 1), where the slices of the pyramid are scaled. They are given by the e.g.f.s exp(t^m / m) * exp(t*(x + y)), which provide a sequence of bivariate polynomials in x and y, whose monomials are to be ordered in degree-lexicographic order. For m = 1 one gets A109649 (resp. A046816), for m = 2 one gets A344911 (resp. A344678), and for m = 3 the current triangle. The row sums have an unexpected interpretation in A336614 (see the link).
%H A344912 mjqxxxx, <a href="https://math.stackexchange.com/q/4164050">Proof of conjectured formulas for A336614</a>, Mathematics Stack Exchange.
%e A344912 Triangle begins:
%e A344912 [0] 1;
%e A344912 [1] 1, 1;
%e A344912 [2] 1, 2,  1;
%e A344912 [3] 1, 3,  3,  1,  2;
%e A344912 [4] 1, 4,  6,  4,  1,  8,  8;
%e A344912 [5] 1, 5, 10, 10,  5,  1, 20, 40,  20;
%e A344912 [6] 1, 6, 15, 20, 15,  6,  1, 40, 120, 120,  40,  40;
%e A344912 [7] 1, 7, 21, 35, 35, 21,  7,  1,  70, 280, 420, 280, 70, 280, 280.
%e A344912 .
%e A344912 p_{6}(x, y) = x^6 + 6*x^5*y + 15*x^4*y^2 + 20*x^3*y^3 + 15*x^2*y^4 + 6*x*y^5 + y^6 + 40*x^3 + 120*x^2*y + 120*x*y^2 + 40*y^3 + 40.
%p A344912 B := (n, k) -> n!/(k!*(n - 3*k)!*(3^k)): C := n -> seq(binomial(n, j), j=0..n):
%p A344912 T := (n, k) -> B(n, k)*C(n - 3*k): seq(seq(T(n, k), k = 0..n/3), n = 0..8);
%t A344912 gf := Exp[t^3 / 3] Exp[t (x + y)]; ser := Series[gf, {t, 0, 9}];
%t A344912 P[n_] := Expand[n! Coefficient[ser, t, n]];
%t A344912 DegLexList[p_] := MonomialList[p, {x, y}, "DegreeLexicographic"] /. x->1 /. y->1;
%t A344912 Table[DegLexList[P[n]], {n, 0, 7}] // Flatten
%Y A344912 m=1: A109649, (A046816) [row sums A000244], scaling A007318 [row sums A000079].
%Y A344912 m=2: A344911, (A344678) [row sums A005425], scaling A100861 [row sums A000085].
%Y A344912 m=3: this triangle      [row sums A336614], scaling A118931 [row sums A001470].
%K A344912 nonn,tabf
%O A344912 0,5
%A A344912 _Peter Luschny_, Jun 04 2021