A372873: Triangle read by rows: T(n,k) is the number of flattened Catalan words of length n with exactly k runs of descents.

A372873 ↗ nonnegative coefficients

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

Recurrence from A372873

\(P_{n}(x) = \left(4\,x\right)\,P_{n-1}(x) + \left(x - 4\,x^{2}\right)\,P_{n-2}(x)\)

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

\(P_{1}(x) = 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	0	2							2
2	0	1	4						5
3	0	0	6	8					14
4	0	0	1	24	16				41
5	0	0	0	10	80	32			122
6	0	0	0	1	60	240	64		365
7	0	0	0	0	14	280	672	128	1094

All 15 rows b-files: A007051 (row sums), A372868 (reversed triangle), A372873 ?

Technical details

Node id: n-5576233c61c40153 · revision 0 · JSON
Definition source: oeis_catalog:A372873
Fingerprint: f6-aa2cb32670ed1336

Observed on the stored rows ?

Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 ?

A372873 triangle 66 terms, 11 rows · tabl · nonn · catalog b-file?
A372868 reversed triangle 56 terms, 14 rows · tabf · nonn · catalog b-file?

Row sums ? A124302, A007051

Submission text ?

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

%e Example

Triangle begins:
  0: 1;
  1: 0, 2;
  2: 0, 1, 4;
  3: 0, 0, 6,  8;
  4: 0, 0, 1, 24, 16;
  5: 0, 0, 0, 10, 80,  32;
  6: 0, 0, 0,  1, 60, 240,   64;
  7: 0, 0, 0,  0, 14, 280,  672,  128;
  8: 0, 0, 0,  0,  1, 112, 1120, 1792,  256;
  9: 0, 0, 0,  0,  0,  18,  672, 4032, 4608, 512;

%F Formula

The row polynomials satisfy P(n,x) = 4*x*P(n-1,x) + (x-4*x^2)*P(n-2,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[4*x*P[n - 1, x] + (x - 4*x^2)*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] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = 4*x*v[i - 1] + (x - 4*x^2)*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

A372868 reversed triangle, offset 1, from P₀, low zeros dropped

%e Example

Triangle begins:
   1:   1;
   2:   2;
   3:   4,    1;
   4:   8,    6;
   5:  16,   24,    1;
   6:  32,   80,   10;
   7:  64,  240,   60,   1;
   8: 128,  672,  280,  14;
   9: 256, 1792, 1120, 112,  1;
  10: 512, 4608, 4032, 672, 18;

%F Formula

Row n lists the coefficients of P(n,x) in descending powers of x, down to the lowest nonzero coefficient, where P(n,x) = 4*x*P(n-1,x) + (x-4*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[4*x*P[n - 1, x] + (x - 4*x^2)*P[n - 2, x]];
Table[Reverse[CoefficientList[Expand[P[n, x]/x^Exponent[P[n, x], x, Min]], x]], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = 4*x*v[i - 1] + (x - 4*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p/x^valuation(p, x))); c = Vecrev(c); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

A372873 triangle, offset 1, from P₀

%e Example

Triangle begins:
   1: 1;
   2: 0, 2;
   3: 0, 1, 4;
   4: 0, 0, 6,  8;
   5: 0, 0, 1, 24, 16;
   6: 0, 0, 0, 10, 80,  32;
   7: 0, 0, 0,  1, 60, 240,   64;
   8: 0, 0, 0,  0, 14, 280,  672,  128;
   9: 0, 0, 0,  0,  1, 112, 1120, 1792,  256;
  10: 0, 0, 0,  0,  0,  18,  672, 4032, 4608, 512;

%F Formula

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

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[4*x*P[n - 1, x] + (x - 4*x^2)*P[n - 2, x]];
Table[PadRight[CoefficientList[P[n, x], x], n], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2*x; for(n = 3, N, my(i = n); v[i] = 4*x*v[i - 1] + (x - 4*x^2)*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c = concat(c, vector(n - #c)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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

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

Related ?

Same fingerprint ?

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