A272867 → basis A013611 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-9 - 4\,x + 8\,x^{2} + 12\,n + 16\,n x\right)\,P_{n}(x) = \left(18 + 144\,x + 384\,x^{2} + 384\,x^{3} + 128\,x^{4} + 72\,n + 384\,n x + 576\,n x^{2} + 256\,n x^{3}\right)\,P_{n-1}(x)\)

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

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	30	56	16																											102
2	324	1824	3072	1792	256																									7268
3	2808	26784	95040	156672	124416	43008	4096																							452824
4	22032	297216	1631232	4700160	7686144	7225344	3735552	917504	65536																					26280720
5	163296	2851200	21254400	88473600	226160640	368492544	384860160	251658240	96337920	18350080	1048576																			1459650656
6	1166400	25007616	235892736	1288949760	4523765760	10696458240	17397448704	19520815104	14898954240	7455375360	2264924160	352321536	16777216																	78677856832
7	8118144	206406144	2356687872	15969042432	71526260736	223343640576	499898843136	812371673088	960087195648	817694244864	490695819264	198709346304	50029658112	6576668672	268435456															4149742040448
8	55427328	1630347264	21857402880	176913580032	965500010496	3758995537920	10776679022592	23158283304960	37619087966208	46241597620224	42743648747520	29273691783168	14459275837440	4915590070272	1056561954816	120259084288	4294967296													215293922664704
9	372874752	12456032256	191798710272	1805923123200	11632724803584	54346246520832	190528902660096	511703443832832	1065593251823616	1730738625970176	2193947778613248	2160933338087424	1636790024798208	936352262651904	392770464251904	115036404056064	21569325760512	2164663517184	68719476736											11026186727564800
10	2479113216	92714803200	1612431360000	17317512806400	128652673351680	701853962600448	2913693566238720	9413824327188480	24011370387210240	48768501027962880	79209609012707328	102926778661601280	106650096547921920	87463615958876160	56052415589253120	27514110253989888	10026257555128320	2579041961902080	428809534832640	38482906972160	1099511627776									558847238577447936
11	16325867520	675769982976	13100359827456	158069871083520	1330823447838720	8307607551344640	39889456975577088	150854695668154368	456464165461032960	1116585641115648000	2222193104230809600	3609545085265379328	4786769749289730048	5169702948506173440	4521574280252620800	3172889566453432320	1760978201003163648	756668646511607808	243691634011668480	55877180923576320	8345293254819840	677299162710016	17592186044416							28082534833598093312
12	106662334464	4841163915264	103510353641472	1386360743657472	13047319865327616	91748407298752512	500480201491218432	2170489921943371776	7609791674125910016	21818508831797280768	51555952004947771392	100887938839873585152	163893165358973976576	221077426099124699136	247141156443062796288	227978544400325148672	172313068599521574912	105589258313212624896	51657074976265076736	19731756507230896128	5692505948531195904	1177339458836496384	159596311794941952	11821949021847552	281474976710656					1401072447861144752128
13	692216782848	34184189804544	799140331192320	11764794541473792	122407471285272576	957633025636564992	5851765016969084928	28643164221069066240	114281484122163511296	376270764345328140288	1031264978831092482048	2366876954303523717120	4566219914464985088000	7418895245695858507776	10153089270681622806528	11685943686402637037568	11273742582708248248320	9066217831094833643520	6028760563762977570816	3277169665958266011648	1432980609408090243072	492580716662438756352	128684026165135933440	24238373194508009472	3007841601130070016	204913783045357568	4503599627370496			69476021495331751747584
14	4466757353472	238436029956096	6050131410419712	97110265288458240	1106955603645825024	9537597230001684480	64552210601355509760	352132986255892807680	1576280889395842646016	5865462697922786230272	18314059913359395913728	48307246710358303309824	108149639172740830199808	206122895987666445139968	334945488217980473966592	464115649854975799984128	547699451620463732588544	548969327522982572064768	465321889377714551390208	331446438384269034258432	196655243703283114573824	96024168520999671693312	37948598223518904090624	11856516400787434242048	2829961800747910692864	489334113912313872384	55988750567470006272	3530822107858468864	72057594037927936	3427124677896338865274880

Technical details

Node id: n-e3c47c78f24a0470 · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-86cc915b987fb8f3

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:        30,          56,           16;
  2:       324,        1824,         3072,          1792,            256;
  3:      2808,       26784,        95040,        156672,         124416,          43008,            4096;
  4:     22032,      297216,      1631232,       4700160,        7686144,        7225344,         3735552,          917504,            65536;
  5:    163296,     2851200,     21254400,      88473600,      226160640,      368492544,       384860160,       251658240,         96337920,         18350080,          1048576;
  6:   1166400,    25007616,    235892736,    1288949760,     4523765760,    10696458240,     17397448704,     19520815104,      14898954240,       7455375360,       2264924160,        352321536,         16777216;
  7:   8118144,   206406144,   2356687872,   15969042432,    71526260736,   223343640576,    499898843136,    812371673088,     960087195648,     817694244864,     490695819264,     198709346304,      50029658112,      6576668672,       268435456;
  8:  55427328,  1630347264,  21857402880,  176913580032,   965500010496,  3758995537920,  10776679022592,  23158283304960,   37619087966208,   46241597620224,   42743648747520,   29273691783168,   14459275837440,   4915590070272,   1056561954816,    120259084288,     4294967296;
  9: 372874752, 12456032256, 191798710272, 1805923123200, 11632724803584, 54346246520832, 190528902660096, 511703443832832, 1065593251823616, 1730738625970176, 2193947778613248, 2160933338087424, 1636790024798208, 936352262651904, 392770464251904, 115036404056064, 21569325760512, 2164663517184, 68719476736;

%F Formula

The row polynomials satisfy (12*n-9+(16*n-4)*x+8*x^2)*P(n,x) = (72*n+18+(384*n+144)*x+(576*n+384)*x^2+(256*n+384)*x^3+128*x^4)*P(n-1,x) for n >= 1, with P(0,x) = 1.

%t Mathematica

P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((72*n + 18 + (384*n + 144)*x + (576*n + 384)*x^2 + (256*n + 384)*x^3 + 128*x^4)*P[n - 1, x])/(12*n - 9 + (16*n - 4)*x + 8*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((72*n + 18 + (384*n + 144)*x + (576*n + 384)*x^2 + (256*n + 384)*x^3 + 128*x^4)*v[i - 1])/(12*n - 9 + (16*n - 4)*x + 8*x^2)); 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 basis_map_A013611 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    ↑ Iterated real-rootedness preservers (above)
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Parent-row provers of polytool (above)

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.