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).

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, 15, 6, 1, 40, 120, 120, 40, 40, 1, 7, 21, 35, 35, 21, 7, 1, 70, 280, 420, 280, 70, 280, 280, 1, 8, 28, 56, 70, 56, 28, 8, 1, 112, 560, 1120, 1120, 560, 112, 1120, 2240, 1120

Offset: 0

Author

Peter Luschny, Jun 04 2021

Keywords

• nonn
• tabf

Comments

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).

Examples

			Triangle begins:
[0] 1;
[1] 1, 1;
[2] 1, 2,  1;
[3] 1, 3,  3,  1,  2;
[4] 1, 4,  6,  4,  1,  8,  8;
[5] 1, 5, 10, 10,  5,  1, 20, 40,  20;
[6] 1, 6, 15, 20, 15,  6,  1, 40, 120, 120,  40,  40;
[7] 1, 7, 21, 35, 35, 21,  7,  1,  70, 280, 420, 280, 70, 280, 280.
.
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.
		

Links

• mjqxxxx, Proof of conjectured formulas for A336614, Mathematics Stack Exchange.

Crossrefs

m=1: A109649, (A046816) [row sums A000244], scaling A007318 [row sums A000079].

m=2: A344911, (A344678) [row sums A005425], scaling A100861 [row sums A000085].

m=3: this triangle [row sums A336614], scaling A118931 [row sums A001470].

Programs

Maple

B := (n, k) -> n!/(k!*(n - 3*k)!*(3^k)): C := n -> seq(binomial(n, j), j=0..n):
T := (n, k) -> B(n, k)*C(n - 3*k): seq(seq(T(n, k), k = 0..n/3), n = 0..8);

Mathematica

gf := Exp[t^3 / 3] Exp[t (x + y)]; ser := Series[gf, {t, 0, 9}];
P[n_] := Expand[n! Coefficient[ser, t, n]];
DegLexList[p_] := MonomialList[p, {x, y}, "DegreeLexicographic"] /. x->1 /. y->1;
Table[DegLexList[P[n]], {n, 0, 7}] // Flatten