A225473 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-21 - 36\,x - 4\,x^{2} + 16\,x^{3}\right)\,P_{n-2}(x) + \left(-12\,x - 12\,x^{2} + 16\,x^{3} + 16\,x^{4}\right)\,P_{n-2}'(x) + \left(10 + 12\,x - 4\,x^{2}\right)\,P_{n-1}(x) + \left(4\,x + 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 7 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
1	1								1
2	7	4							11
3	49	104	32						185
4	343	1660	1824	384					4211
5	2401	22160	51392	38400	6144				120497
6	16807	271924	1106880	1678080	952320	122880			4148891
7	117649	3190904	20821472	52999680	59566080	27279360	2949120		166924265
8	823543	36505900	362441184	1411127424	2590586880	2311249920	887685120	82575360	7682995331

All 15 rows

Technical details

Node id: n-29f6988ef49acca3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-6787cdb08a7f183b

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1:        1;
   2:        7,          4;
   3:       49,        104,          32;
   4:      343,       1660,        1824,          384;
   5:     2401,      22160,       51392,        38400,          6144;
   6:    16807,     271924,     1106880,      1678080,        952320,        122880;
   7:   117649,    3190904,    20821472,     52999680,      59566080,      27279360,       2949120;
   8:   823543,   36505900,   362441184,   1411127424,    2590586880,    2311249920,     887685120,      82575360;
   9:  5764801,  411429920,  6014141312,  33917506560,   92531625984,  132347658240,   97851801600,   32369541120,    2642411520;
  10: 40353607, 4594854244, 96748932480, 763092764160, 2926750740480, 6100311490560, 7133272473600, 4501678325760, 1307993702400, 95126814720;

%F Formula

The row polynomials satisfy P(n,x) = -(21+36*x+4*x^2-16*x^3)*P(n-2,x) - (12*x+12*x^2-16*x^3-16*x^4)*d/dx P(n-2,x) + (10+12*x-4*x^2)*P(n-1,x) + (4*x+4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 7 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(21 + 36*x + 4*x^2 - 16*x^3)*P[n - 2, x] - (12*x + 12*x^2 - 16*x^3 - 16*x^4)*D[P[n - 2, x], x] + (10 + 12*x - 4*x^2)*P[n - 1, x] + (4*x + 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 7 + 4*x; for(n = 3, N, my(i = n); v[i] = -(21 + 36*x + 4*x^2 - 16*x^3)*v[i - 2] - (12*x + 12*x^2 - 16*x^3 - 16*x^4)*deriv(v[i - 2], x) + (10 + 12*x - 4*x^2)*v[i - 1] + (4*x + 4*x^2)*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, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= …?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.