A075504 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 18\,x + 9\,n x\right)\,P_{n-1}(x) + \left(-9\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 27\,x + 81\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	27	81															109
1	1	54	567	729														1351
2	1	90	2025	10935	6561													19612
3	1	135	5265	65610	203391	59049												333451
4	1	189	11340	255150	1974861	3720087	531441											6493069
5	1	252	21546	765450	11160261	57041334	67493007	4782969										141264820
6	1	324	37422	1928934	45605511	458810730	1607609025	1219657095	43046721									3376695763
7	1	405	60750	4286520	149767947	2511058725	18124795305	44625100770	21996874431	387420489								87799365343
8	1	495	93555	8660520	419818707	10598527863	131122437930	697117731750	1226874595221	396331160247	3486784401							2465959810690
9	1	594	138105	16238475	1043376147	37047106404	703442942532	6597627438600	26323112938221	33521945231214	7137447668847	31381059609						74353064138749
10	1	702	196911	28667925	2358692622	112170188988	3037410645984	44583546335328	323216347675221	981154011007170	912229968911625	128505439098855	282429536481					2393123710957813
11	1	819	272727	48162114	4938805872	303224291370	11113664253120	235940417032320	2730727849782933	15525889656392115	36233774365169745	24758714599712730	2313380333315871	2541865828329				81812390963020066
12	1	945	368550	77616630	9706855158	747716819850	35674831854090	1036124243256960	17594974122819093	162985193544670497	734898808902814920	1329174591745823550	670798674525559581	41643387865514007	22876792454961			2958191064076428793
13	1	1080	487620	120736980	18089451198	1708695480492	102969345640590	3925785623438250	92195919637320213	1271468563282273356	9536099260315021758	34399620992372494950	48521083977375207381	18153207600055622694	749603858371707087	205891132094649		112727516544416978299
14	1	1224	633420	182177100	32215677858	3662356209876	272130198209298	13193026731091350	410184555135818463	7909574777169328692	89638618747098243186	549348981049383669882	1596504028634137480131	1764912230785563088410	490886209059873519825	13493075341822822215	1853020188851841	4513118224822056822772

Technical details

Node id: n-a1d96d78bb1f4642 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-f08b49fd82dd971b

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,  27,     81;
  1: 1,  54,    567,      729;
  2: 1,  90,   2025,    10935,       6561;
  3: 1, 135,   5265,    65610,     203391,       59049;
  4: 1, 189,  11340,   255150,    1974861,     3720087,       531441;
  5: 1, 252,  21546,   765450,   11160261,    57041334,     67493007,       4782969;
  6: 1, 324,  37422,  1928934,   45605511,   458810730,   1607609025,    1219657095,       43046721;
  7: 1, 405,  60750,  4286520,  149767947,  2511058725,  18124795305,   44625100770,    21996874431,      387420489;
  8: 1, 495,  93555,  8660520,  419818707, 10598527863, 131122437930,  697117731750,  1226874595221,   396331160247,    3486784401;
  9: 1, 594, 138105, 16238475, 1043376147, 37047106404, 703442942532, 6597627438600, 26323112938221, 33521945231214, 7137447668847, 31381059609;

%F Formula

The row polynomials satisfy P(n,x) = (1+(9*n+18)*x)*P(n-1,x) - 9*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 27*x + 81*x^2.

%t Mathematica

P[0, x_] := 1 + 27*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (9*n + 18)*x)*P[n - 1, x] - 9*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 + 27*x + 81*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (9*n + 18)*x)*v[i - 1] - 9*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 9).?
  - 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 9).

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.