A359363 → D

nonnegative coefficients

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

Recurrence

\(\left(-6\,n - 5\,n^{2} + 5\,n^{3} + 5\,n^{4} + n^{5}\right)\,P_{n}(x) = \left(-6\,n + 6\,n x - 5\,n^{2} + n^{2} x + 5\,n^{3} - 22\,n^{3} x + 5\,n^{4} + 5\,n^{4} x + n^{5} + 10\,n^{5} x\right)\,P_{n-1}(x) + \left(-9\,x - 9\,x^{2} - 6\,n x + 3\,n x^{2} + 27\,n^{2} x + 30\,n^{2} x^{2} + 21\,n^{3} x - 9\,n^{3} x^{2} + 3\,n^{4} x - 15\,n^{4} x^{2}\right)\,P_{n-1}'(x) + \left(-3\,x^{2} - 3\,x^{3} - n x^{2} - n x^{3} + 10\,n^{2} x^{2} + 10\,n^{2} x^{3} + 6\,n^{3} x^{2} + 6\,n^{3} x^{3}\right)\,P_{n-1}''(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	P_n(1)
0								0
1	1							1
2	1	2						3
3	1	8	3					12
4	1	20	30	4				55
5	1	40	150	80	5			276
6	1	70	525	700	175	6		1477
7	1	112	1470	3920	2450	336	7	8296

All 15 rows

Technical details

Node id: n-c00581c65a074f69 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-f7f5c9aa0a83f4e6

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 0;
  1: 1;
  2: 1,   2;
  3: 1,   8,    3;
  4: 1,  20,   30,     4;
  5: 1,  40,  150,    80,      5;
  6: 1,  70,  525,   700,    175,     6;
  7: 1, 112, 1470,  3920,   2450,   336,     7;
  8: 1, 168, 3528, 16464,  20580,  7056,   588,   8;
  9: 1, 240, 7560, 56448, 123480, 84672, 17640, 960, 9;

%F Formula

The row polynomials satisfy (n^5+5*n^4+5*n^3-5*n^2-6*n)*P(n,x) = (n^5+5*n^4+5*n^3-5*n^2-6*n+(10*n^5+5*n^4-22*n^3+n^2+6*n)*x)*P(n-1,x) + ((3*n^4+21*n^3+27*n^2-6*n-9)*x-(15*n^4+9*n^3-30*n^2-3*n+9)*x^2)*d/dx P(n-1,x) + ((6*n^3+10*n^2-n-3)*x^2+(6*n^3+10*n^2-n-3)*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (10*n^5 + 5*n^4 - 22*n^3 + n^2 + 6*n)*x)*P[n - 1, x] + ((3*n^4 + 21*n^3 + 27*n^2 - 6*n - 9)*x - (15*n^4 + 9*n^3 - 30*n^2 - 3*n + 9)*x^2)*D[P[n - 1, x], x] + ((6*n^3 + 10*n^2 - n - 3)*x^2 + (6*n^3 + 10*n^2 - n - 3)*x^3)*D[P[n - 1, x], {x, 2}])/(n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n + (10*n^5 + 5*n^4 - 22*n^3 + n^2 + 6*n)*x)*v[i - 1] + ((3*n^4 + 21*n^3 + 27*n^2 - 6*n - 9)*x - (15*n^4 + 9*n^3 - 30*n^2 - 3*n + 9)*x^2)*deriv(v[i - 1], x) + ((6*n^3 + 10*n^2 - n - 3)*x^2 + (6*n^3 + 10*n^2 - n - 3)*x^3)*deriv(deriv(v[i - 1], x), x))/(n^5 + 5*n^4 + 5*n^3 - 5*n^2 - 6*n)); 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

Properties

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows. The coefficients of the Baxter polynomials p(0, x) = 1 and p(n, x) = x*hypergeom([-1 - n, -n, 1 …?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.