A049029 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-56 + 16\,x - x^{2} + 60\,n - 8\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(8\,x - x^{2} - 4\,n x\right)\,P_{n-2}'(x) + \left(-11 + 2\,x + 8\,n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 5 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	5	2														7
3	45	30	3													78
4	585	510	90	4												1189
5	9945	10350	2475	200	5											22975
6	208845	247590	72450	8100	375	6										537366
7	5221125	6855030	2327535	323400	21000	630	7									14748727
8	151412625	216093150	82436130	13452180	1097250	46620	980	8								464538943
9	4996616625	7649992350	3209404275	594193320	57413475	3088260	92610	1440	9							16510802364
10	184874815125	300692942550	136641755250	28046938500	3096694125	198603090	7585200	169200	2025	10						653559505075
11	7579867420125	12998853217350	6326634889575	1416254301000	174409908750	12851775090	588208005	16790400	289575	2750	11					28509476802631
12	341094033905625	613106982838350	316850119636050	76415719634100	10316403405000	851880645000	44992345860	1545338520	34236675	470250	3630	12				1358681712459072
13	16713607661375625	31337537209728750	17079016575696075	4396084247154600	642289030007625	58381238916000	3468439774800	137958780960	3689996310	65315250	731445	4680	13			70230526117482133
14	885821206052908125	1725654224648103750	986352217477877250	268952739941585700	42105580019377875	4156858693137150	272749392115200	12241458028800	380293403490	8149541400	117927810	1097460	5915	14		3913328206245109939
15	50491808745015763125	101859587441695833750	60783262168166913375	17452300686465645000	2904631306109032500	308251934997756750	22032880178584275	1095166904832000	38490711509250	960417958500	16865658810	203275800	1597050	7350	15	233823009797734367550

Technical details

Node id: n-efcf89935f283045 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-81bb74537eeb9ad3

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:            5,            2;
   3:           45,           30,            3;
   4:          585,          510,           90,           4;
   5:         9945,        10350,         2475,         200,          5;
   6:       208845,       247590,        72450,        8100,        375,         6;
   7:      5221125,      6855030,      2327535,      323400,      21000,       630,       7;
   8:    151412625,    216093150,     82436130,    13452180,    1097250,     46620,     980,      8;
   9:   4996616625,   7649992350,   3209404275,   594193320,   57413475,   3088260,   92610,   1440,    9;
  10: 184874815125, 300692942550, 136641755250, 28046938500, 3096694125, 198603090, 7585200, 169200, 2025, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^2-60*n+56+(8*n-16)*x+x^2)*P(n-2,x) - ((4*n-8)*x+x^2)*d/dx P(n-2,x) + (8*n-11+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 5 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 60*n + 56 + (8*n - 16)*x + x^2)*P[n - 2, x] - ((4*n - 8)*x + x^2)*D[P[n - 2, x], x] + (8*n - 11 + 2*x)*P[n - 1, x] + 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] = 5 + 2*x; for(n = 3, N, my(i = n); v[i] = -(16*n^2 - 60*n + 56 + (8*n - 16)*x + x^2)*v[i - 2] - ((4*n - 8)*x + x^2)*deriv(v[i - 2], x) + (8*n - 11 + 2*x)*v[i - 1] + 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 multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on multiplier sequences
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 2
by Laguerre’s theorem on multiplier sequences

Proof chains ? JSON

Laguerre’s theorem on multiplier sequences [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool ? step ›
- from Triangle read by rows, the Bell transform of the quartic factorial numbers A007696(n+1) without column 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.