A257612 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(12 - 40\,x + 60\,x^{2} - 24\,n + 96\,n x - 184\,n x^{2} - 32\,n^{2} x + 144\,n^{2} x^{2} - 32\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(8\,x - 32\,x^{2} + 24\,x^{3} - 16\,n x + 80\,n x^{2} - 64\,n x^{3} - 32\,n^{2} x^{2} + 32\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-20 + 36\,x + 16\,n - 48\,n x + 16\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,x + 12\,x^{2} + 8\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	24	8													32
3	184	368	24												576
4	1216	7360	3648	64											12288
5	7584	107648	161472	30336	160										307200
6	46208	1349504	4521216	2699008	231040	384									8847360
7	278912	15528192	99733632	132978176	38820480	1673472	896								289013760
8	1677312	169785344	1908701184	4787214336	3181168640	509356032	11741184	2048							10569646080
9	10072576	1798492160	33343444992	142158426112	177698032640	66686889984	6294722560	80580608	4608						428070666240
10	60454912	18669856768	547740377088	3714757763072	7818713436160	5572136644608	1278060879872	74679427072	544094208	10240					19025362944000
11	362772480	191293140992	8620527974400	88775254818816	292741322895360	351289587474432	155356695932928	22988074598400	860819134464	3627724800	22528				920827566489600
12	2176729088	1943404298240	131591100678144	1988751854911488	9769381159403520	18267058548670464	13677133623164928	3977503709822976	394773302034432	9717021491200	23944019968	49152			48217879845273600
13	13060579328	19634302058496	1964709880283136	42464815822766080	299448339340615680	826641244003958784	964414784671285248	479117342944985088	95545835601223680	6549032934277120	107988661321728	156726951936	106496		2716273897950412800
14	78363918336	197649078517760	28860705166000128	874390438105645056	8604942217074933760	33710164589200048128	57864887080277114880	44946886118933397504	15488895990734880768	2185976095264112640	105822585608667136	1185894471106560	1018730938368	229376	163812210461009510400

Technical details

Node id: n-9da9c144cb7ef765 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-a7d11e539b3a8882

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:        2;
  2:       24,          8;
  3:      184,        368,          24;
  4:     1216,       7360,        3648,           64;
  5:     7584,     107648,      161472,        30336,          160;
  6:    46208,    1349504,     4521216,      2699008,       231040,         384;
  7:   278912,   15528192,    99733632,    132978176,     38820480,     1673472,        896;
  8:  1677312,  169785344,  1908701184,   4787214336,   3181168640,   509356032,   11741184,     2048;
  9: 10072576, 1798492160, 33343444992, 142158426112, 177698032640, 66686889984, 6294722560, 80580608, 4608;

%F Formula

The row polynomials satisfy (2*n-3)*P(n,x) = -(24*n-12+(32*n^2-96*n+40)*x+(32*n^3-144*n^2+184*n-60)*x^2)*P(n-2,x) - ((16*n-8)*x+(32*n^2-80*n+32)*x^2-(32*n^2-64*n+24)*x^3)*d/dx P(n-2,x) + (16*n-20+(16*n^2-48*n+36)*x)*P(n-1,x) + ((8*n-12)*x-(8*n-12)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(24*n - 12 + (32*n^2 - 96*n + 40)*x + (32*n^3 - 144*n^2 + 184*n - 60)*x^2)*P[n - 2, x] - ((16*n - 8)*x + (32*n^2 - 80*n + 32)*x^2 - (32*n^2 - 64*n + 24)*x^3)*D[P[n - 2, x], x] + (16*n - 20 + (16*n^2 - 48*n + 36)*x)*P[n - 1, x] + ((8*n - 12)*x - (8*n - 12)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(24*n - 12 + (32*n^2 - 96*n + 40)*x + (32*n^3 - 144*n^2 + 184*n - 60)*x^2)*v[i - 2] - ((16*n - 8)*x + (32*n^2 - 80*n + 32)*x^2 - (32*n^2 - 64*n + 24)*x^3)*deriv(v[i - 2], x) + (16*n - 20 + (16*n^2 - 48*n + 36)*x)*v[i - 1] + ((8*n - 12)*x - (8*n - 12)*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 4*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

References

[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.