A395972 → D+1

nonnegative coefficients

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

Recurrence

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

\(P_{1}(x) = 20 + 32\,x + 8\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
1	20	32	8								60
2	80	360	288	48							776
3	272	2560	5760	3072	384						12048
4	832	13600	64000	96000	38400	3840					216672
5	2368	59904	489600	1536000	1728000	552960	46080				4414912
6	6400	232064	2935296	15993600	37632000	33868800	9031680	645120			100344960
7	16640	819200	14852096	125239296	511795200	963379200	722534400	165150720	10321920		2514108672
8	41984	2695680	66355200	802013184	5072191488	16582164480	26011238400	16721510400	3344302080	185794560	68788307456

All 15 rows

Technical details

Node id: n-a60c92d68f2a3294 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-c039e466817b8579

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 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:     20,       32,          8;
   2:     80,      360,        288,          48;
   3:    272,     2560,       5760,        3072,          384;
   4:    832,    13600,      64000,       96000,        38400,          3840;
   5:   2368,    59904,     489600,     1536000,      1728000,        552960,         46080;
   6:   6400,   232064,    2935296,    15993600,     37632000,      33868800,       9031680,         645120;
   7:  16640,   819200,   14852096,   125239296,    511795200,     963379200,     722534400,      165150720,       10321920;
   8:  41984,  2695680,   66355200,   802013184,   5072191488,   16582164480,   26011238400,    16721510400,     3344302080,      185794560;
   9: 103424,  8396800,  269568000,  4423680000,  40100659200,  202887659520,  552738816000,   743178240000,   418037760000,    74317824000,    3715891200;
  10: 249856, 25028608, 1016012800, 21745152000, 267632640000, 1940871905280, 8183135600640, 19108970496000, 22481141760000, 11240570880000, 1798491340800, 81749606400;

%F Formula

The row polynomials satisfy (n^2+1+(2*n-1)*x+x^2)*P(n,x) = (2*n^2+4*n+4+(4*n^3+6*n^2+8*n-2)*x+(8*n^2+4*n+2)*x^2+(4*n+2)*x^3)*P(n-1,x) - ((2*n^2+4*n+4)*x^2+(4*n+2)*x^3+2*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 20 + 32*x + 8*x^2.

%t Mathematica

P[1, x_] := 20 + 32*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 4*n + 4 + (4*n^3 + 6*n^2 + 8*n - 2)*x + (8*n^2 + 4*n + 2)*x^2 + (4*n + 2)*x^3)*P[n - 1, x] - ((2*n^2 + 4*n + 4)*x^2 + (4*n + 2)*x^3 + 2*x^4)*D[P[n - 1, x], x])/(n^2 + 1 + (2*n - 1)*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 20 + 32*x + 8*x^2; for(n = 2, N, my(i = n); v[i] = ((2*n^2 + 4*n + 4 + (4*n^3 + 6*n^2 + 8*n - 2)*x + (8*n^2 + 4*n + 2)*x^2 + (4*n + 2)*x^3)*v[i - 1] - ((2*n^2 + 4*n + 4)*x^2 + (4*n + 2)*x^3 + 2*x^4)*deriv(v[i - 1], x))/(n^2 + 1 + (2*n - 1)*x + x^2)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) = 2^n * [x^k] hypergeom([-n, -n, 1], [1], x).?
  - 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
  - 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

Related ?

Same fingerprint ?

A144084 → D+1

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.