A225469 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-21 + 6\,x + 3\,x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 12\,x^{2}\right)\,P_{n-2}'(x) + \left(10 - 2\,x\right)\,P_{n-1}(x) + \left(4\,x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	4	1														5
3	19	12	1													32
4	106	121	24	1												252
5	661	1200	418	40	1											2320
6	4384	12121	6540	1070	60	1										24176
7	29959	125292	98251	24320	2285	84	1									280192
8	207526	1320481	1454964	510111	71190	4319	112	1								3568704
9	1446121	14116800	21472228	10249680	1989246	176736	7476	144	1							49458432
10	10103164	152412241	317184360	201082540	52098480	6348342	388920	12108	180	1						739630336
11	70663099	1656387372	4696684501	3891732640	1310040670	210106008	17463642	780480	18615	220	1					11853877248
12	494464546	18079112041	69727979904	74735811821	32079816780	6585448738	712668264	42840402	1456290	27445	264	1				202459626496
13	3460720381	197881834800	1037621573638	1428772836920	771753630791	198802636032	27232823332	2113130448	95939547	2561680	39094	312	1			3667737726976
14	24223448344	2169780336361	15470572970580	27243894724410	18342629405100	5843668013183	992911639720	96679538956	5626455120	199480567	4291716	54106	364	1		70190190358528
15	169559355439	23819141586252	231010037171551	518706779004960	432290898430755	168482782640172	34975242024723	4186688631840	303607320445	13717964260	389962573	6901440	73073	420	1	1413958851067904

Technical details

Node id: n-945797c7dad86c3c · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-29750a34cb2a3058

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:        4,         1;
   3:       19,        12,         1;
   4:      106,       121,        24,         1;
   5:      661,      1200,       418,        40,        1;
   6:     4384,     12121,      6540,      1070,       60,       1;
   7:    29959,    125292,     98251,     24320,     2285,      84,      1;
   8:   207526,   1320481,   1454964,    510111,    71190,    4319,    112,     1;
   9:  1446121,  14116800,  21472228,  10249680,  1989246,  176736,   7476,   144,   1;
  10: 10103164, 152412241, 317184360, 201082540, 52098480, 6348342, 388920, 12108, 180, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(21-6*x-3*x^2)*P(n-2,x) - (12*x-12*x^2)*d/dx P(n-2,x) + (10-2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 4 + x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[-(21 - 6*x - 3*x^2)*P[n - 2, x] - (12*x - 12*x^2)*D[P[n - 2, x], x] + (10 - 2*x)*P[n - 1, x] + 4*x*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] = 4 + x; for(n = 3, N, my(i = n); v[i] = -(21 - 6*x - 3*x^2)*v[i - 2] - (12*x - 12*x^2)*deriv(v[i - 2], x) + (10 - 2*x)*v[i - 1] + 4*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, 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, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.?
  - 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.