A241171 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 30\,x - 90\,x^{2}\right)\,P_{n-2}(x) + \left(-5\,x - 39\,x^{2} - 78\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} - 8\,x^{3} - 12\,x^{4}\right)\,P_{n-2}''(x) + \left(5 + 30\,x\right)\,P_{n-1}(x) + \left(5\,x + 15\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)

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

\(P_{1}(x) = 6\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	6														6
2	30	180													210
3	126	2520	7560												10206
4	510	26460	226800	453600											707370
5	2046	253440	4740120	24948000	37422000										67365606
6	8190	2342340	86486400	908107200	3405402000	4086482400									8488828530
7	32766	21326760	1482160680	27891864000	197513316000	572107536000	572107536000								1371123772206
8	131070	192923820	24610294800	786226240800	9411558156000	49601723371200	116709937344000	100037089152000							276571337613690
9	524286	1740246480	401867517240	21132273708000	403679758482000	3458226595824000	14413677261984000	28510570408320000	21382927806240000						68190617712846006
10	2097150	15677946900	6502970628000	552418893734400	16275883886262000	213534932821610400	1406627068182336000	4829690627169408000	8232427205402400000	5488284803601600000					20187399458608022850
11	8388606	141164436600	104706003817800	14200650581040000	631506378638520000	12240538228638720000	119919022958694960000	637163731008604800000	1855589092097700960000	2777072110622409600000	1666243266373445760000				7068873573165891003006
12	33554430	1270731587580	1681224967422000	361298624755068000	23905783303922520000	668746869750623664000	9388492821365898240000	72578959940474507520000	321752656712859372000000	812293592357054808000000	1083058123142739744000000	590758976259676224000000			2890523816868345587576010
13	134217726	11437590921120	26952970205430360	9133339116922020000	890415335483503830000	35379108154568898720000	694488102332504800320000	7520540785657272172800000	47689018358562363182400000	180870706564837537248000000	402572704372156362844800000	483831601556674827456000000	241915800778337413728000000		1365131139202351858687459206
14	536870910	102942344821860	431727902105572800	229950656135376321600	32808452554552204530000	1830809654208674655516000	49400313690919953041280000	730600008960379885367040000	6376417428790478245176000000	34125536873043015223824000000	112882815942675192558144000000	224497863122297119939584000000	245544537790012474933920000000	113328248210774988431040000000	737537283538733819377319953170

Technical details

Node id: n-59467d32d5647a53 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-c1fcefc8c9c99a70

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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:      0;
  1:      6;
  2:     30,        180;
  3:    126,       2520,         7560;
  4:    510,      26460,       226800,         453600;
  5:   2046,     253440,      4740120,       24948000,        37422000;
  6:   8190,    2342340,     86486400,      908107200,      3405402000,       4086482400;
  7:  32766,   21326760,   1482160680,    27891864000,    197513316000,     572107536000,      572107536000;
  8: 131070,  192923820,  24610294800,   786226240800,   9411558156000,   49601723371200,   116709937344000,   100037089152000;
  9: 524286, 1740246480, 401867517240, 21132273708000, 403679758482000, 3458226595824000, 14413677261984000, 28510570408320000, 21382927806240000;

%F Formula

The row polynomials satisfy P(n,x) = -(4+30*x+90*x^2)*P(n-2,x) - (5*x+39*x^2+78*x^3)*d/dx P(n-2,x) - (x^2+8*x^3+12*x^4)*d^2/dx^2 P(n-2,x) + (5+30*x)*P(n-1,x) + (5*x+15*x^2)*d/dx P(n-1,x) + (x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 6.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 6;
P[n_, x_] := P[n, x] = Expand[-(4 + 30*x + 90*x^2)*P[n - 2, x] - (5*x + 39*x^2 + 78*x^3)*D[P[n - 2, x], x] - (x^2 + 8*x^3 + 12*x^4)*D[P[n - 2, x], {x, 2}] + (5 + 30*x)*P[n - 1, x] + (5*x + 15*x^2)*D[P[n - 1, x], x] + (x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 6; for(n = 2, N, my(i = n + 1); v[i] = -(4 + 30*x + 90*x^2)*v[i - 2] - (5*x + 39*x^2 + 78*x^3)*deriv(v[i - 2], x) - (x^2 + 8*x^3 + 12*x^4)*deriv(deriv(v[i - 2], x), x) + (5 + 30*x)*v[i - 1] + (5*x + 15*x^2)*deriv(v[i - 1], x) + (x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), 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: Joffe's central differences of zero, T(n,k), 1 <= k <= n.?
  - 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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.