A176230 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 3\,x + 6\,x^{2} + 4\,n x - 10\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-4\,x^{2} + 14\,x^{3} - 8\,n x^{3}\right)\,P_{n-1}'(x) + \left(4\,x^{4}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	1	1							2
2	1	6	3						10
3	1	15	45	15					76
4	1	28	210	420	105				764
5	1	45	630	3150	4725	945			9496
6	1	66	1485	13860	51975	62370	10395		140152
7	1	91	3003	45045	315315	945945	945945	135135	2390480

All 15 rows

Technical details

Node id: n-01922c46a12adf16 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-86230d7253d1861d

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: 1,   1;
  2: 1,   6,    3;
  3: 1,  15,   45,     15;
  4: 1,  28,  210,    420,     105;
  5: 1,  45,  630,   3150,    4725,      945;
  6: 1,  66, 1485,  13860,   51975,    62370,     10395;
  7: 1,  91, 3003,  45045,  315315,   945945,    945945,    135135;
  8: 1, 120, 5460, 120120, 1351350,  7567560,  18918900,  16216200,   2027025;
  9: 1, 153, 9180, 278460, 4594590, 41351310, 192972780, 413513100, 310134825, 34459425;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n-3)*x+(4*n^2-10*n+6)*x^2)*P(n-1,x) - (4*x^2+(8*n-14)*x^3)*d/dx P(n-1,x) + 4*x^4*d^2/dx^2 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[(1 + (4*n - 3)*x + (4*n^2 - 10*n + 6)*x^2)*P[n - 1, x] - (4*x^2 + (8*n - 14)*x^3)*D[P[n - 1, x], x] + 4*x^4*D[P[n - 1, x], {x, 2}]];
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] = (1 + (4*n - 3)*x + (4*n^2 - 10*n + 6)*x^2)*v[i - 1] - (4*x^2 + (8*n - 14)*x^3)*deriv(v[i - 1], x) + 4*x^4*deriv(deriv(v[i - 1], x), 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) 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

Related ?

Same fingerprint ?

A176230: Exponential Riordan array [1/sqrt(1-2x), x/(1-2x)].

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.