A075503 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-128 + 48\,x + 7\,x^{2}\right)\,P_{n-2}(x) + \left(-64\,x + 56\,x^{2}\right)\,P_{n-2}'(x) + \left(24 - 6\,x\right)\,P_{n-1}(x) + \left(8\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 88 + 26\,x + x^{2}\)

\(P_{1}(x) = 960 + 544\,x + 51\,x^{2} + x^{3}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	88	26	1								115
1	960	544	51	1							1556
2	11776	10880	1840	84	1						24581
3	159744	219136	58560	4640	125	1					442206
4	2326528	4530176	1770496	215040	9800	174	1				8852215
5	35389440	96600064	52555776	9117696	622720	18368	231	1			194304296
6	551550976	2120744960	1556807680	368492544	35245056	1532160	31584	296	1		4634405257
7	8707375104	47706013696	46387691520	14514257920	1860956160	111562752	3346560	50880	369	1	119291254962

All 15 rows

Technical details

Node id: n-d741cd559e72192f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c9f198fd808e3a9a

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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:            88,             26,              1;
  1:           960,            544,             51,              1;
  2:         11776,          10880,           1840,             84,             1;
  3:        159744,         219136,          58560,           4640,           125,            1;
  4:       2326528,        4530176,        1770496,         215040,          9800,          174,           1;
  5:      35389440,       96600064,       52555776,        9117696,        622720,        18368,         231,         1;
  6:     551550976,     2120744960,     1556807680,      368492544,      35245056,      1532160,       31584,       296,        1;
  7:    8707375104,    47706013696,    46387691520,    14514257920,    1860956160,    111562752,     3346560,     50880,      369,      1;
  8:  138378477568,  1093639602176,  1395147145216,   564375060480,   94085939200,   7453974528,   304668672,   6673920,    77880,    450,   1;
  9: 2206539448320, 25424595779584, 42391176216576, 21830530236416, 4627748618240, 470338502656, 25118134272, 742612992, 12386880, 114400, 539, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(128-48*x-7*x^2)*P(n-2,x) - (64*x-56*x^2)*d/dx P(n-2,x) + (24-6*x)*P(n-1,x) + 8*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 88 + 26*x + x^2, P(1,x) = 960 + 544*x + 51*x^2 + x^3.

%t Mathematica

P[0, x_] := 88 + 26*x + x^2;
P[1, x_] := 960 + 544*x + 51*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(128 - 48*x - 7*x^2)*P[n - 2, x] - (64*x - 56*x^2)*D[P[n - 2, x], x] + (24 - 6*x)*P[n - 1, x] + 8*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 88 + 26*x + x^2; v[2] = 960 + 544*x + 51*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(128 - 48*x - 7*x^2)*v[i - 2] - (64*x - 56*x^2)*deriv(v[i - 2], x) + (24 - 6*x)*v[i - 1] + 8*x*deriv(v[i - 1], 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Stirling2 triangle with scaled diagonals (powers of 8).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? 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
[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.