A038331 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(27 + 10\,x + 10\,n\right)\,P_{n}(x) = \left(629 + 540\,x + 100\,x^{2} + 170\,n + 100\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 629 + 540\,x + 100\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	629	540	100								1269
1	13583	18870	8100	1000							41553
2	280041	543320	377400	108000	10000						1318761
3	5595907	14002050	13583000	6290000	1350000	100000					40920957
4	109328989	335754420	420061500	271660000	94350000	16200000	1000000				1248354909
5	2099968503	7653029230	11751404700	9801435000	4754050000	1320900000	189000000	10000000			37579787433
6	39802851281	167997480240	306121169200	313370792000	196028700000	76064800000	17612000000	2160000000	100000000		1119257792721
7	746406046187	3582256615290	7559886610800	9183635076000	7050842820000	3528516600000	1140972000000	226440000000	24300000000	1000000000	33044255768277

All 15 rows

Technical details

Node id: n-99d7cc44f38135a6 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c94624b66e9d9245

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): yes

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:             629,              540,              100;
  1:           13583,            18870,             8100,             1000;
  2:          280041,           543320,           377400,           108000,            10000;
  3:         5595907,         14002050,         13583000,          6290000,          1350000,           100000;
  4:       109328989,        335754420,        420061500,        271660000,         94350000,         16200000,          1000000;
  5:      2099968503,       7653029230,      11751404700,       9801435000,       4754050000,       1320900000,        189000000,        10000000;
  6:     39802851281,     167997480240,     306121169200,     313370792000,     196028700000,      76064800000,      17612000000,      2160000000,       100000000;
  7:    746406046187,    3582256615290,    7559886610800,    9183635076000,    7050842820000,    3528516600000,    1140972000000,    226440000000,     24300000000,     1000000000;
  8:  13874781550149,   74640604618700,  179112830764500,  251996220360000,  229590876900000,  141016856400000,   58808610000000,  16299600000000,   2830500000000,   270000000000,   10000000000;
  9: 256031225357023, 1526225970516390, 4105233254028500, 6567470461365000, 6929896059900000, 5050999291800000, 2585309034000000, 924135300000000, 224119500000000, 34595000000000, 2970000000000, 100000000000;

%F Formula

The row polynomials satisfy (10*n+27+10*x)*P(n,x) = (170*n+629+(100*n+540)*x+100*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 629 + 540*x + 100*x^2.

%t Mathematica

P[0, x_] := 629 + 540*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((170*n + 629 + (100*n + 540)*x + 100*x^2)*P[n - 1, x])/(10*n + 27 + 10*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 629 + 540*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((170*n + 629 + (100*n + 540)*x + 100*x^2)*v[i - 1])/(10*n + 27 + 10*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 basis_map_A013609 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*5^j.?
    
    Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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
    
    Parent-row provers of polytool polytool ? 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.