A176230 → D+1

nonnegative coefficients

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

Recurrence

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	2	1							3
2	9	8	1						18
3	60	75	18	1					154
4	525	840	294	32	1				1692
5	5670	11025	5040	810	50	1			22596
6	72765	166320	93555	19800	1815	72	1		354328
7	1081080	2837835	1891890	495495	60060	3549	98	1	6370008

All 15 rows

Technical details

Node id: n-066e40bc69f26be1 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-973eb94a4c84f80a

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 ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:         1;
  1:         2,          1;
  2:         9,          8,         1;
  3:        60,         75,        18,         1;
  4:       525,        840,       294,        32,        1;
  5:      5670,      11025,      5040,       810,       50,       1;
  6:     72765,     166320,     93555,     19800,     1815,      72,      1;
  7:   1081080,    2837835,   1891890,    495495,    60060,    3549,     98,     1;
  8:  18243225,   54054000,  41621580,  12972960,  1951950,  152880,   6300,   128,   1;
  9: 344594250, 1137161025, 992431440, 358378020, 64324260, 6265350, 342720, 10404, 162, 1;

%F Formula

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

%t Mathematica

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

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((2*n^2 + n - 1 - n*x - x^2)*v[i - 1] + ((2*n - 2)*x - 2*x^2)*deriv(v[i - 1], x))/(n - x)); 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 Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - 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

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[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.