basis-map coefficients of A371080, shifted → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-36 + 12\,x - x^{2} + 36\,n - 6\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	0	2														2
2	0	6	3													9
3	0	36	27	4												67
4	0	324	297	72	5											698
5	0	3888	4050	1260	150	6										9354
6	0	58320	66582	24300	3825	270	7									153304
7	0	1049760	1285956	526176	99225	9450	441	8								2971016
8	0	22044960	28579716	12764304	2741445	317520	20286	672	9							66468912
9	0	529079040	718980624	344449584	81750060	10910214	857304	39312	972	10						1686067120
10	0	14285134080	20206095408	10258779600	2637813600	392675850	35875791	2041200	70470	1350	11					47818487360
11	0	428554022400	627610563360	334704848544	91957882500	14945651820	1534395555	102236904	4410450	118800	1815	12				1499414132160
12	0	14142282739200	21353979624480	11882074086432	3452991183180	603555969060	68071647105	5127412752	260561367	8820900	190575	2376	13			51508352237440
13	0	509122178611200	789956690590080	456226639944192	139160275202520	25871604305976	3154727926350	262383027192	15148548558	607065030	16563690	293436	3042	14		1923770272081280
14	0	19855764965836800	31571994200929920	18846114545276928	5997534032828520	1175984898176088	153217927484622	13838341404888	885974299353	40507256790	1313755443	29513484	436527	3822	15	77615376737203200

Technical details

Node id: n-6e04084af2055336 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-b88ed3a3c932c924

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: 0,         2;
  2: 0,         6,         3;
  3: 0,        36,        27,         4;
  4: 0,       324,       297,        72,        5;
  5: 0,      3888,      4050,      1260,      150,        6;
  6: 0,     58320,     66582,     24300,     3825,      270,      7;
  7: 0,   1049760,   1285956,    526176,    99225,     9450,    441,     8;
  8: 0,  22044960,  28579716,  12764304,  2741445,   317520,  20286,   672,   9;
  9: 0, 529079040, 718980624, 344449584, 81750060, 10910214, 857304, 39312, 972, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(9*n^2-36*n+36+(6*n-12)*x+x^2)*P(n-2,x) + (6*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 2*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 36*n + 36 + (6*n - 12)*x + x^2)*P[n - 2, x] + (6*n - 9 + 2*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 36*n + 36 + (6*n - 12)*x + x^2)*v[i - 2] + (6*n - 9 + 2*x)*v[i - 1]); 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 Laguerre’s theorem on multiplier sequences
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on multiplier sequences
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from 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 basis-map coefficients of Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1…?
  - Parent-row provers of polytool polytool ? step › direct check

References

[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.