A285061 → reverse

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	4	1														5
2	16	24	1													41
3	64	240	124	1												429
4	256	1792	2656	624	1											5329
5	1024	11520	33920	26400	3124	1										75989
6	4096	67584	331520	546560	250096	15624	1									1215481
7	16384	372736	2745344	7822080	8105664	2313360	78124	1								21453693
8	65536	1966080	20299776	88940544	165398016	114627072	21132736	390624	1							412820385
9	262144	10027008	138215424	863256576	2529343488	3270274560	1574682880	191757120	1953124	1						8579772325
10	1048576	49807360	883752960	7461273600	31698788352	66197311488	61893399040	21237905920	1733626576	9765624	1					191166679497
11	4194304	242221056	5377884160	59008942080	343172087808	1057258954752	1637717137408	1137139407360	283027283264	15641701680	48828124	1				4538638641997
12	16777216	1157627904	31442599936	435017482240	3319983439872	14181026365440	32982342418432	38940617515008	20463479058176	3741921489152	140970627616	244140624	1			114238219541617
13	67108864	5452595200	177863655424	3029216526336	29375580602368	166283558977536	543179134271488	980321030520832	899606884047872	362846829945600	49208861869440	1269712211040	1220703124	1		3035305413035125
14	268435456	25367150592	978631786496	20120730599424	241700199989248	1752030718197760	7660073983344640	19673479015956480	28106453299212288	20343131884787712	6365231556552960	644794053146880	11432292711856	6103515624	1	84819458105387417

Technical details

Node id: n-fdb150e76bed506a · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-7d794259d02ffafb

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 0)
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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      1;
  1:      4,        1;
  2:     16,       24,         1;
  3:     64,      240,       124,         1;
  4:    256,     1792,      2656,       624,          1;
  5:   1024,    11520,     33920,     26400,       3124,          1;
  6:   4096,    67584,    331520,    546560,     250096,      15624,          1;
  7:  16384,   372736,   2745344,   7822080,    8105664,    2313360,      78124,         1;
  8:  65536,  1966080,  20299776,  88940544,  165398016,  114627072,   21132736,    390624,       1;
  9: 262144, 10027008, 138215424, 863256576, 2529343488, 3270274560, 1574682880, 191757120, 1953124, 1;

%F Formula

The row polynomials satisfy P(n,x) = (4+(4*n-3)*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(4 + (4*n - 3)*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4 + (4*n - 3)*x)*v[i - 1] - 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 - 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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
- from Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 1).?
  - 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 ?

A285061: Sheffer triangle S2[4,1] = (exp(x), exp(4*x) - 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.