A286785 → D

nonnegative coefficients

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

Recurrence

\(\left(56 + 86\,n + 34\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(392 + 112\,x + 387\,n + 186\,n x + 130\,n^{2} + 86\,n^{2} x + 15\,n^{3} + 12\,n^{3} x\right)\,P_{n-1}(x) + \left(2 + 68\,x - 70\,x^{2} + 3\,n + 45\,n x - 48\,n x^{2} + n^{2} + 7\,n^{2} x - 8\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 14 + 4\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	P_n(1)
0	14	4								18
1	72	54	6							132
2	330	440	132	8						910
3	1430	2860	1560	260	10					6120
4	6006	16380	13650	4200	450	12				40698
5	24752	86632	99960	47600	9520	714	14			269192
6	100776	434112	651168	434112	135660	19152	1064	16		1776060
7	406980	2093040	3907008	3418632	1508220	335160	35280	1512	18	11705850

All 15 rows

Technical details

Node id: n-24fe2fab32454333 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-0fc3a9782387e944

Observed on the stored rows ?

Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:      14,        4;
  1:      72,       54,         6;
  2:     330,      440,       132,         8;
  3:    1430,     2860,      1560,       260,        10;
  4:    6006,    16380,     13650,      4200,       450,       12;
  5:   24752,    86632,     99960,     47600,      9520,      714,       14;
  6:  100776,   434112,    651168,    434112,    135660,    19152,     1064,      16;
  7:  406980,  2093040,   3907008,   3418632,   1508220,   335160,    35280,    1512,    18;
  8: 1634380,  9806280,  22064130,  24227280,  14132580,  4462920,   743820,   60720,  2070,   20;
  9: 6537520, 44945450, 118973250, 158631000, 116886000, 49092120, 11688600, 1518000, 99000, 2750, 22;

%F Formula

The row polynomials satisfy (4*n^3+34*n^2+86*n+56)*P(n,x) = (15*n^3+130*n^2+387*n+392+(12*n^3+86*n^2+186*n+112)*x)*P(n-1,x) + (n^2+3*n+2+(7*n^2+45*n+68)*x-(8*n^2+48*n+70)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 14 + 4*x.

%t Mathematica

P[0, x_] := 14 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((15*n^3 + 130*n^2 + 387*n + 392 + (12*n^3 + 86*n^2 + 186*n + 112)*x)*P[n - 1, x] + (n^2 + 3*n + 2 + (7*n^2 + 45*n + 68)*x - (8*n^2 + 48*n + 70)*x^2)*D[P[n - 1, x], x])/(4*n^3 + 34*n^2 + 86*n + 56)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 14 + 4*x; for(n = 1, N, my(i = n + 1); v[i] = ((15*n^3 + 130*n^2 + 387*n + 392 + (12*n^3 + 86*n^2 + 186*n + 112)*x)*v[i - 1] + (n^2 + 3*n + 2 + (7*n^2 + 45*n + 68)*x - (8*n^2 + 48*n + 70)*x^2)*deriv(v[i - 1], x))/(4*n^3 + 34*n^2 + 86*n + 56)); 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 T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.?
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? 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.