A322944 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-10 + 10\,x - x^{2} + 7\,n - 2\,n x - n^{2}\right)\,P_{n-2}(x) + \left(15\,x - 3\,x^{2} - 3\,n x\right)\,P_{n-2}'(x) + \left(-6 + 2\,x + 2\,n\right)\,P_{n-1}(x) + \left(3\,x\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	2														3
3	2	12	3													17
4	6	76	45	4												131
5	24	544	564	112	5											1249
6	120	4400	7020	2320	225	6										14091
7	720	39840	90840	44160	6950	396	7									182913
8	5040	399840	1240680	827680	187250	17052	637	8								2678187
9	40320	4408320	17969280	15724800	4779600	616896	36456	960	9							43576641
10	362880	52980480	276151680	307005440	120027600	20541024	1704024	70464	1377	10						778844979
11	3628800	689472000	4497897600	6201305600	3024364000	657558720	71677200	4131840	126090	1900	11					15150163761
12	39916800	9659865600	77498467200	130023308800	77312004000	20725763520	2845790640	214135680	9061470	212300	2541	12				318328528563
13	479001600	144977817600	1409462208000	2833820774400	2018017232000	652370019840	109553776640	10318809600	567115560	18348000	340252	3312	13			7179585446817
14	6227020800	2320603084800	26997248678400	64223339980800	53992706768000	20687981233920	4157265432320	476043842560	32591936520	1364048400	34813636	523536	4225	14		172895407367931
15	87178291200	39462706483200	543425878195200	1513133151129600	1484089550944000	664742703905280	157168471940480	21412694983680	1774042910640	92139247200	3032253224	62584704	778414	5292	15	4425391613652129

Technical details

Node id: n-5d0f3e585742a565 · revision 0 · JSON
Definition source: derived:theta_plus_one
Fingerprint: f6-260aff1e54e87d47

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:      1,        2;
   3:      2,       12,         3;
   4:      6,       76,        45,         4;
   5:     24,      544,       564,       112,         5;
   6:    120,     4400,      7020,      2320,       225,        6;
   7:    720,    39840,     90840,     44160,      6950,      396,       7;
   8:   5040,   399840,   1240680,    827680,    187250,    17052,     637,     8;
   9:  40320,  4408320,  17969280,  15724800,   4779600,   616896,   36456,   960,    9;
  10: 362880, 52980480, 276151680, 307005440, 120027600, 20541024, 1704024, 70464, 1377, 10;

%F Formula

The row polynomials satisfy P(n,x) = -(n^2-7*n+10+(2*n-10)*x+x^2)*P(n-2,x) - ((3*n-15)*x+3*x^2)*d/dx P(n-2,x) + (2*n-6+2*x)*P(n-1,x) + 3*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.

%t Mathematica

P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 10 + (2*n - 10)*x + x^2)*P[n - 2, x] - ((3*n - 15)*x + 3*x^2)*D[P[n - 2, x], x] + (2*n - 6 + 2*x)*P[n - 1, x] + 3*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] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = -(n^2 - 7*n + 10 + (2*n - 10)*x + x^2)*v[i - 2] - ((3*n - 15)*x + 3*x^2)*deriv(v[i - 2], x) + (2*n - 6 + 2*x)*v[i - 1] + 3*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 Coefficients of a family of orthogonal polynomials. Triangle read by rows, T(n, k) for 0 <= k <= n.?
  - 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.