A075499 → reverse

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1 + 12\,x + 16\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	12	16															29
1	1	24	112	64														201
2	1	40	400	960	256													1657
3	1	60	1040	5760	7936	1024												15821
4	1	84	2240	22400	77056	64512	4096											170389
5	1	112	4256	67200	435456	989184	520192	16384										2032785
6	1	144	7392	169344	1779456	7956480	12390400	4177920	65536									26546673
7	1	180	12000	376320	5843712	43545600	139694080	152862720	33488896	262144								376085653
8	1	220	18480	760320	16380672	183794688	1010606080	2387968000	1867841536	268173312	1048576							5736591885
9	1	264	27280	1425600	40710912	642453504	5421678592	22600089600	40075329536	22682271744	2146435072	4194304						93614616409
10	1	312	38896	2516800	92032512	1945202688	23410376704	152720375808	492077121536	663887544320	274333696000	17175674880	16777216					1625661357673
11	1	364	53872	4228224	192704512	5258373120	85656862720	808210923520	4157366140928	10505429975040	10896534405120	3309180026880	137422176256	67108864				29905322979421
12	1	420	72800	6814080	378746368	12966553600	274958295040	3549230530560	26787271999488	110282217357312	221005133905920	177653730508800	39847582498816	1099444518912	268435456			580513190237573
13	1	480	96320	10599680	705822208	29631393792	793620439040	13447729152000	140362648977408	860325833342976	2867778350481408	4597756408627200	2882307270639616	479270434504704	8795824586752	1073741824		11850869542405409
14	1	544	125120	15993600	1257005568	63510859776	2097401765888	45192546713600	624480898449408	5351930600620032	26956901684084736	73424436820180992	94837435443183616	46596186764738560	5760041038643200	70367670435840	4294967296	253669139947767777

Technical details

Node id: n-9ada29ffbad64b45 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-6d39ca603c4714ce

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0: 1,  12,    16;
  1: 1,  24,   112,      64;
  2: 1,  40,   400,     960,      256;
  3: 1,  60,  1040,    5760,     7936,      1024;
  4: 1,  84,  2240,   22400,    77056,     64512,       4096;
  5: 1, 112,  4256,   67200,   435456,    989184,     520192,       16384;
  6: 1, 144,  7392,  169344,  1779456,   7956480,   12390400,     4177920,       65536;
  7: 1, 180, 12000,  376320,  5843712,  43545600,  139694080,   152862720,    33488896,      262144;
  8: 1, 220, 18480,  760320, 16380672, 183794688, 1010606080,  2387968000,  1867841536,   268173312,    1048576;
  9: 1, 264, 27280, 1425600, 40710912, 642453504, 5421678592, 22600089600, 40075329536, 22682271744, 2146435072, 4194304;

%F Formula

The row polynomials satisfy P(n,x) = (1+(4*n+8)*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 12*x + 16*x^2.

%t Mathematica

P[0, x_] := 1 + 12*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (4*n + 8)*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 + 12*x + 16*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (4*n + 8)*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 Stirling2 triangle with scaled diagonals (powers of 4).?
  - 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 ?

Stirling2 triangle with scaled diagonals (powers of 4).

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.