A049029 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-56 + 28\,x + 9\,x^{2} + 60\,n - 16\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(14\,x - 28\,x^{2} + 4\,x^{3} - 8\,n x + 16\,n x^{2}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(-11 + 4\,x + 8\,n\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{2}(x) = 5 + 4\,x\)

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	4														9
3	45	60	9													114
4	585	1020	270	16												1891
5	9945	20700	7425	800	25											38895
6	208845	495180	217350	32400	1875	36										955686
7	5221125	13710060	6982605	1293600	105000	3780	49									27316219
8	151412625	432186300	247308390	53808720	5486250	279720	6860	64								890488929
9	4996616625	15299984700	9628212825	2376773280	287067375	18529560	648270	11520	81							32607844236
10	184874815125	601385885100	409925265750	112187754000	15483470625	1191618540	53096400	1353600	18225	100						1325103277465
11	7579867420125	25997706434700	18979904668725	5665017204000	872049543750	77110650540	4117456035	134323200	2606175	27500	121					59175910334871
12	341094033905625	1226213965676700	950550358908150	305662878536400	51582017025000	5111283870000	314946421020	12362708160	308130075	4702500	39930	144				2880542159923704
13	16713607661375625	62675074419457500	51237049727088225	17584336988618400	3211445150038125	350287433496000	24279078423600	1103670247680	33209966790	653152500	8045895	56160	169			151797217999966669
14	885821206052908125	3451308449296207500	2959056652433631750	1075810959766342800	210527900096889375	24941152158822900	1909245744806400	97931664230400	3422640631410	81495414000	1297205910	13169520	76895	196		8609477002660337181
15	50491808745015763125	203719174883391667500	182349786504500740125	69809202745862580000	14523156530545162500	1849511609986540500	154230161250089925	8761335238656000	346416403583250	9604179585000	185522246910	2439309600	20761650	102900	225	522905988724356789210

Technical details

Node id: n-f4134c9d170b148d · revision 0 · JSON
Definition source: derived:theta_plus_one_squared
Fingerprint: f6-73ec16ecdb9f69d1

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,            4;
   3:           45,           60,            9;
   4:          585,         1020,          270,           16;
   5:         9945,        20700,         7425,          800,          25;
   6:       208845,       495180,       217350,        32400,        1875,         36;
   7:      5221125,     13710060,      6982605,      1293600,      105000,       3780,       49;
   8:    151412625,    432186300,    247308390,     53808720,     5486250,     279720,     6860,      64;
   9:   4996616625,  15299984700,   9628212825,   2376773280,   287067375,   18529560,   648270,   11520,    81;
  10: 184874815125, 601385885100, 409925265750, 112187754000, 15483470625, 1191618540, 53096400, 1353600, 18225, 100;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^2-60*n+56+(16*n-28)*x-9*x^2)*P(n-2,x) - ((8*n-14)*x-(16*n-28)*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (8*n-11+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 5 + 4*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 5 + 4*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 60*n + 56 + (16*n - 28)*x - 9*x^2)*P[n - 2, x] - ((8*n - 14)*x - (16*n - 28)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (8*n - 11 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*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 + 4*x; for(n = 3, N, my(i = n); v[i] = -(16*n^2 - 60*n + 56 + (16*n - 28)*x - 9*x^2)*v[i - 2] - ((8*n - 14)*x - (16*n - 28)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (8*n - 11 + 4*x)*v[i - 1] + (2*x - 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, 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.