A316649 → basis A013609

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + n^{2}\right)\,P_{n}(x) = \left(-8 - 16\,x - 8\,x^{2} + 12\,n + 24\,n x + 12\,n x^{2} - 4\,n^{2} - 8\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-2}(x) + \left(12 + 28\,x + 16\,x^{2} - 30\,n - 70\,n x - 40\,n x^{2} + 12\,n^{2} + 28\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 3 + 10\,x + 8\,x^{2}\)

\(P_{2}(x) = 26 + 150\,x + 316\,x^{2} + 288\,x^{3} + 96\,x^{4}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	x¹⁷	x¹⁸	x¹⁹	x²⁰	x²¹	x²²	x²³	x²⁴	x²⁵	x²⁶	x²⁷	x²⁸	P_n(1)
0	1																													1
1	3	10	8																											21
2	26	150	316	288	96																									876
3	252	2064	6924	12184	11872	6080	1280																							40656
4	2568	27240	124872	323256	517136	523840	328320	116480	17920																					1981632
5	26928	350688	2035872	6940288	15391152	23209760	24112384	17047296	7852544	2128896	258048																			99353856
6	287648	4440800	31184128	131739840	373001120	745866592	1080379840	1142491392	875642880	474485760	172591104	37847040	3784704																	5073742848
7	3112896	55575040	457737536	2305765248	7937500736	19757480704	36678187200	51598574464	55286807552	44908406784	27226933248	11949527040	3589521408	660701184	56229888															262472060928
8	34013312	689376384	6513848704	38094173824	154352456832	459534900608	1040018086016	1825476675456	2511768453376	2718687971328	2307460571136	1519763976192	761578954752	280741773312	71805566976	11386552320	843448320													13708750798848
9	374416128	8492659200	90533993472	602755424256	2808892310016	9730580923392	25972322718720	54620986011648	91762344678144	124082812639744	135388294150144	118930125332480	83494291234816	46216232910848	19728499867648	6267477032960	1395532103680	194367979520	12745441280											721307661828096
10	4145895936	104049133056	1235147572224	9222183991296	48577386568704	191904238064640	590004709429248	1445733682993152	2867858398960128	4651217694316032	6201893230754816	6811318662184960	6151287482081280	4543585917911040	2718348395642880	1297145412845568	482153633415168	134555535409152	26524537847808	3293422026752	193730707456									38176161597751296
11	46127840256	1269038592000	16599981874176	137356016424960	806852737013760	3579360701743104	12453795022510080	34837444739026944	79676144243297280	150690719818240000	237425828151364608	312948704806492160	345567114431766528	319271498455203840	245810567675412480	156570868741767168	81586207292129280	34205225334079488	11257395368427520	2800883717898240	495298971435008	55477429862400	2958796259328							2030197617598660608
12	515268544512	15419790796800	220352517629952	2001231690602496	12968783997345792	63825624382722048	247887798125383680	779359604783198208	2018757737900918784	4361707517794142208	7928267531526752256	12192917680883251200	15916763427124023296	17656142518191177728	16626794196485963776	13250627629686194176	8887407492319150080	4974557944265834496	2295436877793067008	858063767476371456	253344261504761856	56861899945410560	9116544407699456	930048290848768	45368209309696					108394021774361100288
13	5775088103424	186766960386048	2894770550857728	28626499386114048	202854406673485824	1096715702259572736	4702751179415764992	16414632611891576832	47492342323432968192	115395307318844399616	237684777289203965952	417778863064704712704	629435574752129101824	814950084665901850624	907537346436289789952	868597105287612137472	712726444995198582784	499224023959471652864	296550162774311829504	148012850616333238272	61275255325832773632	20663682767492481024	5531442825914220544	1130748524179226624	165806845577920512	15529887032934400	697972450918400			5806616715344140566528
14	64912164888576	2255986702540800	37693526869377024	403162520953700352	3101038420959977472	18268463093178015744	85715597939625738240	328862504122285326336	1051097562262438699008	2836744252354522718208	6529417366669636927488	12911787691361546158080	22051491442436403863552	32641521490527453536256	41964411603895009263616	46889623403801524502528	45508189496919494492160	38287847343049785475072	27828418400830872354816	17383172195677189963776	9264893485996371345408	4171886779766008709120	1565918526770597855232	480962058926021410816	117750154074464976896	22098530696114995200	2985377738103193600	258449227540070400	10768717814169600	311946877162221654245376

Technical details

Node id: n-03c7727a6c6d39ac · revision 0 · JSON
Definition source: derived:basis_map_A013609
Fingerprint: f6-ad0a1921d2e79f69

Observed on the stored rows ?

Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:         3,         10,           8;
  2:        26,        150,         316,          288,            96;
  3:       252,       2064,        6924,        12184,         11872,          6080,           1280;
  4:      2568,      27240,      124872,       323256,        517136,        523840,         328320,         116480,          17920;
  5:     26928,     350688,     2035872,      6940288,      15391152,      23209760,       24112384,       17047296,        7852544,         2128896,          258048;
  6:    287648,    4440800,    31184128,    131739840,     373001120,     745866592,     1080379840,     1142491392,      875642880,       474485760,       172591104,        37847040,        3784704;
  7:   3112896,   55575040,   457737536,   2305765248,    7937500736,   19757480704,    36678187200,    51598574464,    55286807552,     44908406784,     27226933248,     11949527040,     3589521408,      660701184,       56229888;
  8:  34013312,  689376384,  6513848704,  38094173824,  154352456832,  459534900608,  1040018086016,  1825476675456,  2511768453376,   2718687971328,   2307460571136,   1519763976192,   761578954752,   280741773312,    71805566976,   11386552320,     843448320;
  9: 374416128, 8492659200, 90533993472, 602755424256, 2808892310016, 9730580923392, 25972322718720, 54620986011648, 91762344678144, 124082812639744, 135388294150144, 118930125332480, 83494291234816, 46216232910848, 19728499867648, 6267477032960, 1395532103680, 194367979520, 12745441280;

%F Formula

The row polynomials satisfy (n^2-2*n)*P(n,x) = -(4*n^2-12*n+8+(8*n^2-24*n+16)*x+(4*n^2-12*n+8)*x^2)*P(n-2,x) + (12*n^2-30*n+12+(28*n^2-70*n+28)*x+(16*n^2-40*n+16)*x^2)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 3 + 10*x + 8*x^2, P(2,x) = 26 + 150*x + 316*x^2 + 288*x^3 + 96*x^4.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 3 + 10*x + 8*x^2;
P[2, x_] := 26 + 150*x + 316*x^2 + 288*x^3 + 96*x^4;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 12*n + 8 + (8*n^2 - 24*n + 16)*x + (4*n^2 - 12*n + 8)*x^2)*P[n - 2, x] + (12*n^2 - 30*n + 12 + (28*n^2 - 70*n + 28)*x + (16*n^2 - 40*n + 16)*x^2)*P[n - 1, x])/(n^2 - 2*n)]];
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] = 3 + 10*x + 8*x^2; v[3] = 26 + 150*x + 316*x^2 + 288*x^3 + 96*x^4; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^2 - 12*n + 8 + (8*n^2 - 24*n + 16)*x + (4*n^2 - 12*n + 8)*x^2)*v[i - 2] + (12*n^2 - 30*n + 12 + (28*n^2 - 70*n + 28)*x + (16*n^2 - 40*n + 16)*x^2)*v[i - 1])/(n^2 - 2*n)); 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 Geometric basis map (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Geometric basis map (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Geometric basis map (elementary)

Proof chains ? JSON

Geometric basis map (elementary) polytool ? step ›
- from Triangle read by rows in which T(n,k) is the number of length k chains from (0,0) to (n,n) of the poset [n] X [n] order…?
  - Liu–Wang theorem for three-term recurrences [LW07, Thm 2.1, Thm 2.3, Cor 2.4, Prop 3.3] polytool ? step › direct check

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