A038284 → basis A013611 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 12\,x + 12\,n\right)\,P_{n}(x) = \left(434 + 912\,x + 288\,x^{2} + 168\,n + 288\,n x\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 868 + 1824\,x + 576\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	P_n(1)
0	868	1824	576								3268
1	16856	62496	65664	13824							158840
2	301840	1618176	2999808	2101248	331776						7352848
3	5147744	36220800	97090560	119992320	63037440	7962624					329451488
4	84976192	741275136	2607897600	4660346880	4319723520	1815478272	191102976				14420800576
5	1370375552	14276000256	62267111424	146042265600	195734568960	145142710272	50833391616	4586471424			620252895104
6	21715181824	263112105984	1370496024576	3985095131136	7010028748800	7516207448064	4644566728704	1394287312896	110075314176		26315583996160
7	339431482880	4690479273984	28416107446272	98675713769472	215195137081344	302833241948160	270583468130304	143318059057152	37645757448192	2641807540224	1104339203177984

All 15 rows

Technical details

Node id: n-48f947c3236c9f1b · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-29ed0a4e339957b1

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:            868,             1824,               576;
  1:          16856,            62496,             65664,             13824;
  2:         301840,          1618176,           2999808,           2101248,             331776;
  3:        5147744,         36220800,          97090560,         119992320,           63037440,            7962624;
  4:       84976192,        741275136,        2607897600,        4660346880,         4319723520,         1815478272,          191102976;
  5:     1370375552,      14276000256,       62267111424,      146042265600,       195734568960,       145142710272,        50833391616,         4586471424;
  6:    21715181824,     263112105984,     1370496024576,     3985095131136,      7010028748800,      7516207448064,      4644566728704,      1394287312896,       110075314176;
  7:   339431482880,    4690479273984,    28416107446272,    98675713769472,    215195137081344,    302833241948160,    270583468130304,    143318059057152,     37645757448192,      2641807540224;
  8:  5247905883136,   81463555891200,   562857512878080,  2273288595701760,   5920542826168320,  10329366579904512,  12113329677926400,   9277147478753280,   4299541771714560,   1003886865285120,    63403380965376;
  9: 80412794083328, 1385447153147904, 10753189377638400, 49531461133271040, 150037047316316160, 312604661221687296, 454492129515798528, 456845576424652800, 306145866798858240, 126119891970293760, 26502613243527168, 1521681143169024;

%F Formula

The row polynomials satisfy (12*n+19+12*x)*P(n,x) = (168*n+434+(288*n+912)*x+288*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 868 + 1824*x + 576*x^2.

%t Mathematica

P[0, x_] := 868 + 1824*x + 576*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((168*n + 434 + (288*n + 912)*x + 288*x^2)*P[n - 1, x])/(12*n + 19 + 12*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 868 + 1824*x + 576*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((168*n + 434 + (288*n + 912)*x + 288*x^2)*v[i - 1])/(12*n + 19 + 12*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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013611 of Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle whose (i,j)-th entry is binomial(i,j)*8^(i-j)*6^j.?
    
    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
    
    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

Related ?

Same fingerprint ?

A038271 → x ↦ x/(x+1) → D+1

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.