A290319 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-49 + 14\,x - x^{2} + 56\,n - 8\,n x - 16\,n^{2}\right)\,P_{n-2}(x) + \left(-10 + 2\,x + 8\,n\right)\,P_{n-1}(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	2														3
2	5	12	3													20
3	45	118	45	4												212
4	585	1624	762	112	5											3088
5	9945	28778	15390	2920	225	6										57264
6	208845	624228	366357	81840	8375	396	7									1290048
7	5221125	16023390	10095267	2534476	311675	19950	637	8								34206528
8	151412625	475120560	316797828	86960160	12206670	952560	41748	960	9							1043493120
9	4996616625	15981803730	11167009164	3292082384	511520310	46082484	2489004	79392	1377	10						35997684480
10	184874815125	601319971260	437152044663	136696393760	23041354450	2318876280	145856046	5782080	140265	1900	11					1385555235840
11	7579867420125	25023868451910	18825213788073	6187421537044	1115566024650	122723552820	8685453546	403757904	12255705	233750	2541	12				58863762478080
12	341094033905625	1141233815176200	884670423141150	303534254217744	57934748030555	6861239106480	534022887860	28095338304	1005734367	24136200	371470	3312	13			2735891662049280
13	16713607661375625	56602645011445050	45060701456680650	16052739020857656	3218220471269375	405722413854186	34171900462700	1986983448736	80888239575	2300156430	44751850	567528	4225	14		138089878153113600
14	885821206052908125	3033367400929338900	2473121144721242025	910876103381029968	190631608753348945	25365152499795108	2284453540686317	144363723311808	6522433077303	211784112540	4902020123	78899184	838747	5292	15	7521618172800614400

Technical details

Node id: n-677d4fd290d15d11 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-0ef69ca863b1b5c3

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:          1,           2;
  2:          5,          12,           3;
  3:         45,         118,          45,          4;
  4:        585,        1624,         762,        112,         5;
  5:       9945,       28778,       15390,       2920,       225,        6;
  6:     208845,      624228,      366357,      81840,      8375,      396,       7;
  7:    5221125,    16023390,    10095267,    2534476,    311675,    19950,     637,     8;
  8:  151412625,   475120560,   316797828,   86960160,  12206670,   952560,   41748,   960,    9;
  9: 4996616625, 15981803730, 11167009164, 3292082384, 511520310, 46082484, 2489004, 79392, 1377, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(16*n^2-56*n+49+(8*n-14)*x+x^2)*P(n-2,x) + (8*n-10+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(16*n^2 - 56*n + 49 + (8*n - 14)*x + x^2)*P[n - 2, x] + (8*n - 10 + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(16*n^2 - 56*n + 49 + (8*n - 14)*x + x^2)*v[i - 2] + (8*n - 10 + 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
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: T(n, k) is the Sheffer triangle ((1 - 4*x)^(-1/4), (-1/4)*log(1 - 4*x)). A generalized Stirling1…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Parent-row provers of polytool polytool ? 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.