A161198 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-25 + 20\,x - 4\,x^{2} + 20\,n - 8\,n x - 4\,n^{2}\right)\,P_{n-2}(x) + \left(-8 + 4\,x + 4\,n\right)\,P_{n-1}(x)\)

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

\(P_{2}(x) = 3 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	3	2														5
3	11	16	4													31
4	61	118	60	8												247
5	457	1040	728	192	16											2433
6	4323	10978	9320	3440	560	32										28653
7	49443	136160	132076	60160	13840	1536	64									393279
8	663549	1947062	2086420	1101912	316400	49952	4032	128								6169455
9	10223505	31589856	36609072	21619584	7271264	1444352	166656	10240	256							108934785
10	177853635	573867522	708927312	457359328	173508384	40673472	5940480	523776	25344	512						2138679765
11	3448137915	11545978608	15047827284	10436808960	4357775840	1159936512	201086592	22548480	1570560	61440	1024					46221733215
12	73720354365	254948644278	347848899948	256383004328	115826724640	34129365952	6753479040	903032064	80298240	4533760	146432	2048				1090598485095
13	1723066771545	6131201614704	8705399587848	6759413632576	3263245494256	1045666592256	230125584128	35272427520	3754215168	271462400	12683264	344064	4096			27897430413825
14	43709137575075	159539839167330	234621761911992	190624950053200	97422468922064	33508953132512	8052895428864	1378037360128	168721403136	14649130496	878884864	34557952	798720	8192		769042328334525
15	1195958421688275	4466600500656960	6777603805131324	5731286164810752	3078176788641136	1125251614462976	291338243548352	54629032460288	7491532835072	750465171456	54208394240	2743336960	92065792	1835008	16384	22729143615054975

Technical details

Node id: n-07593db20f25af3f · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-42e9f6f4286ebba9

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:         3,         2;
   3:        11,        16,         4;
   4:        61,       118,        60,         8;
   5:       457,      1040,       728,       192,        16;
   6:      4323,     10978,      9320,      3440,       560,       32;
   7:     49443,    136160,    132076,     60160,     13840,     1536,      64;
   8:    663549,   1947062,   2086420,   1101912,    316400,    49952,    4032,    128;
   9:  10223505,  31589856,  36609072,  21619584,   7271264,  1444352,  166656,  10240,   256;
  10: 177853635, 573867522, 708927312, 457359328, 173508384, 40673472, 5940480, 523776, 25344, 512;

%F Formula

The row polynomials satisfy P(n,x) = -(4*n^2-20*n+25+(8*n-20)*x+4*x^2)*P(n-2,x) + (4*n-8+4*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 3 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(4*n^2 - 20*n + 25 + (8*n - 20)*x + 4*x^2)*P[n - 2, x] + (4*n - 8 + 4*x)*P[n - 1, 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] = 3 + 2*x; for(n = 3, N, my(i = n); v[i] = -(4*n^2 - 20*n + 25 + (8*n - 20)*x + 4*x^2)*v[i - 2] + (4*n - 8 + 4*x)*v[i - 1]); 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 of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? 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

Related ?

Same fingerprint ?

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.