A225476 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-3 - 2\,x + x^{2} + x^{3}\right)\,P_{n-2}(x) + \left(-2\,x + x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-2}'(x) + \left(4 + 2\,x - x^{2}\right)\,P_{n-1}(x) + \left(2\,x + x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	2	1														3
2	5	8	2													15
3	14	49	36	6												105
4	41	272	404	192	24											933
5	122	1441	3720	3420	1200	120										10023
6	365	7448	30902	47040	31080	8640	720									126195
7	1094	37969	241836	561666	596400	306600	70560	5040								1821165
8	3281	192032	1824104	6152832	9659664	7822080	3286080	645120	40320							29625513
9	9842	966721	13435920	63730680	140626080	164863440	107533440	38162880	6531840	362880						536223723
10	29525	4853288	97420202	635479680	1905779760	3072746880	2858315040	1557964800	478396800	72576000	3628800					10687190775
11	88574	24325489	698883636	6169924926	24562164000	52633297200	66301139520	50959450080	23830329600	6446563200	878169600	39916800				232544252625
12	265721	121804592	4977191804	58770583872	305260259304	848582127360	1398349719360	1438686627840	940385255040	384797952000	93006144000	11496038400	479001600			5484912970893
13	797162	609554401	35266260120	552060867540	3692925796560	13081628119560	27515780772480	36598412410560	31735077454080	18023540895360	6552901555200	1431176947200	161902540800	6227020800		139387510991823
14	2391485	3049366328	248996065502	5132616390720	43776139319640	194929877862720	513904929698160	862055382988800	957557730729600	716271370214400	359367078470400	117516336537600	23407371187200	2440992153600	87178291200	3796699051667355

Technical details

Node id: n-362d698441b465c3 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-0d549fd1b86bbcc3

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:    2,      1;
  2:    5,      8,        2;
  3:   14,     49,       36,        6;
  4:   41,    272,      404,      192,        24;
  5:  122,   1441,     3720,     3420,      1200,       120;
  6:  365,   7448,    30902,    47040,     31080,      8640,       720;
  7: 1094,  37969,   241836,   561666,    596400,    306600,     70560,     5040;
  8: 3281, 192032,  1824104,  6152832,   9659664,   7822080,   3286080,   645120,   40320;
  9: 9842, 966721, 13435920, 63730680, 140626080, 164863440, 107533440, 38162880, 6531840, 362880;

%F Formula

The row polynomials satisfy P(n,x) = -(3+2*x-x^2-x^3)*P(n-2,x) - (2*x-x^2-3*x^3-x^4)*d/dx P(n-2,x) + (4+2*x-x^2)*P(n-1,x) + (2*x+x^2)*d/dx 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[-(3 + 2*x - x^2 - x^3)*P[n - 2, x] - (2*x - x^2 - 3*x^3 - x^4)*D[P[n - 2, x], x] + (4 + 2*x - x^2)*P[n - 1, x] + (2*x + x^2)*D[P[n - 1, x], 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] = -(3 + 2*x - x^2 - x^3)*v[i - 2] - (2*x - x^2 - 3*x^3 - x^4)*deriv(v[i - 2], x) + (4 + 2*x - x^2)*v[i - 1] + (2*x + 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 - 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 differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k…?
  - Ma–Wang interval theorem [MW08, Thm 2, Cor 3] RealRooted RealRooted.affineEndpointStep_closed ? 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
[MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.