A306364: Triangular array of the number of binary, rooted, leaf-labeled tree topologies with n leaves and k cherries, n >= 2, 1 <= k <= floor(n/2).

A306364 ↗ nonnegative coefficients

Proved for all n all roots real, in (−∞, 0)coefficients of one sign Hadamard kernel interlacing (weak)

Recurrence from A306364

\(P_{n}(x) = \left(1 + 2\,n\right)\,P_{n-1}(x) + \left(1 - x - n^{2} + n^{2} x\right)\,P_{n-2}(x)\)

\(P_{0}(x) = 1\)

\(P_{1}(x) = 3\)

Rows ?

n	x⁰	x¹	x²	x³	P_n(1)
0	1				1
1	3				3
2	12	3			15
3	60	45			105
4	360	540	45		945
5	2520	6300	1575		10395
6	20160	75600	37800	1575	135135
7	181440	952560	793800	99225	2027025

All 15 rows b-files: A306364 ?

Technical details

Node id: n-3951e6b9a140d595 · revision 0 · JSON
Definition source: oeis_catalog:A306364
Fingerprint: f6-125f57a26ac8836e

Observed on the stored rows ?

Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
Palindromic rows: no
γ-positive (gamma_positive): no
Nonnegative coefficients (nonn): yes
Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

A306364 triangle 30 terms, 10 rows · tabf · nonn · catalog b-file?

Row sums ? A001147

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:        1;
  1:        3;
  2:       12,         3;
  3:       60,        45;
  4:      360,       540,        45;
  5:     2520,      6300,      1575;
  6:    20160,     75600,     37800,      1575;
  7:   181440,    952560,    793800,     99225;
  8:  1814400,  12700800,  15876000,   3969000,   99225;
  9: 19958400, 179625600, 314344800, 130977000, 9823275;

%F Formula

The row polynomials satisfy P(n,x) = (2*n+1)*P(n-1,x) - (n^2-1-(n^2-1)*x)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(2*n + 1)*P[n - 1, x] - (n^2 - 1 - (n^2 - 1)*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (2*n + 1)*v[i - 1] - (n^2 - 1 - (n^2 - 1)*x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

A306364 triangle, offset 2, from P₀

%e Example

Triangle begins:
   2:        1;
   3:        3;
   4:       12,         3;
   5:       60,        45;
   6:      360,       540,        45;
   7:     2520,      6300,      1575;
   8:    20160,     75600,     37800,      1575;
   9:   181440,    952560,    793800,     99225;
  10:  1814400,  12700800,  15876000,   3969000,   99225;
  11: 19958400, 179625600, 314344800, 130977000, 9823275;

%F Formula

The row polynomials satisfy P(n,x) = (2*n-3)*P(n-1,x) - (n^2-4*n+3-(n^2-4*n+3)*x)*P(n-2,x) for n >= 4, with P(2,x) = 1, P(3,x) = 3.

%t Mathematica

P[2, x_] := 1;
P[3, x_] := 3;
P[n_, x_] := P[n, x] = Expand[(2*n - 3)*P[n - 1, x] - (n^2 - 4*n + 3 - (n^2 - 4*n + 3)*x)*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 2, 11}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N - 1)); v[1] = 1; v[2] = 3; for(n = 4, N, my(i = n - 1); v[i] = (2*n - 3)*v[i - 1] - (n^2 - 4*n + 3 - (n^2 - 4*n + 3)*x)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i + 1, v[i])))};
T(11)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
by Liu–Wang theorem for three-term recurrences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
by Liu–Wang theorem for three-term recurrences

Proof chains ? JSON

Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

References

[LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003