A272866 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(5\,n + 15\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 15 + 18\,x\)

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	P_n(1)
0																													0
1	15	18																											33
2	150	630	810	324																									1914
3	1125	8100	22275	29160	18225	4374																							83259
4	7500	76500	324000	737100	972000	743580	306180	52488																					3219348
5	46875	618750	3543750	11542500	23540625	31164750	26790750	14434200	4428675	590490																			116701365
6	281250	4556250	32906250	139725000	387281250	735378750	975948750	905418000	575727750	239148450	58458510	6377292																	4061207502
7	1640625	31500000	274640625	1438762500	5049843750	12534243750	22628615625	30082185000	29450688750	20977157250	10571247225	3571283520	725416965	66961566															137404187151
8	9375000	208125000	2126250000	13253625000	56345625000	172991700000	396120375000	688740975000	916735725000	934155180000	722936907000	417393761400	174100071600	49551558840	8609344200	688747536													4553967345576
9	52734375	1328906250	15567187500	112539375000	562127343750	2058103687500	5717911500000	12310240275000	20789861203125	27698040618750	29112337980000	24005721411000	15342568809750	7442778060900	2647373341500	650866421520	98792224695	6973568802											148573184649417
10	292968750	8261718750	109160156250	897960937500	5154257812500	21928092187500	71689176562500	184269785625000	377849015156250	623478092906250	831267833343750	895551158137500	776455230037500	537115461277500	292179618787500	122166594198000	37870352799750	8193943342350	1104148393650	69735688020									4787358172036770
11	1611328125	50273437500	739599609375	6821718750000	44245458984375	214539574218750	807131144531250	2413443346875000	5828343894140625	11487909138750000	18600066866390625	24813883739700000	27276649424578125	24631602798206250	18156299791781250	10810323044730000	5116613970661875	1879376792139000	516189374031375	99722033868600	12081707949465	690383311398							152716725687972963
12	8789062500	300585937500	4872656250000	49805507812500	360228515625000	1961209335937500	8347591054687500	28478225531250000	79208236335937500	181729848248437500	346662617956875000	552503567937937500	737445932744625000	824326591654912500	769904057185312500	597906040673700000	383066851254862500	200115273734392500	83804863078035000	27442736628070500	6765756451700400	1180555462490580	129917586781260	6778308875544					4831401867219508284
13	47607421875	1770996093750	31363769531250	351891210937500	2808085693359375	16962534316406250	80597997949218750	309027200625000000	973204791152343750	2549166550734375000	5603146185578906250	10397861221885312500	16352642306437734375	21835508565959156250	24755718601739531250	23786782918212600000	19297483162801593750	13138377962188050000	7440718006258143750	3461601100663777500	1299689732663697375	383904899885642850	85853871931282650	13658292384221160	1376843990344875	66088511536554			151786541995146218589
14	256347656250	10305175781250	197951660156250	2418609375000000	21108845214843750	140108745410156250	735216220019531250	3130366379765625000	11013491942578125000	32442468613242187500	80782231717687500000	171208151530467187500	310325556725346093750	482513153686662656250	644522310121872656250	739619214611618250000	728097598310764500000	613022505325868137500	439337885462350875000	266176806276643762500	135034229324182938750	56613071195000570250	19260823563600050250	5181339304465833600	1060169872565553750	154977559553219130	14412379246625430	640550188738908	4740410465386874211318

Technical details

Node id: n-a8f6492efcb0f220 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-9de613d7fd734864

Observed on the stored rows ?

Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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:       15,         18;
  2:      150,        630,         810,          324;
  3:     1125,       8100,       22275,        29160,        18225,          4374;
  4:     7500,      76500,      324000,       737100,       972000,        743580,        306180,          52488;
  5:    46875,     618750,     3543750,     11542500,     23540625,      31164750,      26790750,       14434200,        4428675,         590490;
  6:   281250,    4556250,    32906250,    139725000,    387281250,     735378750,     975948750,      905418000,      575727750,      239148450,       58458510,        6377292;
  7:  1640625,   31500000,   274640625,   1438762500,   5049843750,   12534243750,   22628615625,    30082185000,    29450688750,    20977157250,    10571247225,     3571283520,      725416965,      66961566;
  8:  9375000,  208125000,  2126250000,  13253625000,  56345625000,  172991700000,  396120375000,   688740975000,   916735725000,   934155180000,   722936907000,   417393761400,   174100071600,   49551558840,    8609344200,    688747536;
  9: 52734375, 1328906250, 15567187500, 112539375000, 562127343750, 2058103687500, 5717911500000, 12310240275000, 20789861203125, 27698040618750, 29112337980000, 24005721411000, 15342568809750, 7442778060900, 2647373341500, 650866421520, 98792224695, 6973568802;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (5*n+15*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 15 + 18*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 15 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((5*n + 15*n*x + 9*n*x^2)*P[n - 1, x])/(n - 1)]];
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] = 15 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = ((5*n + 15*n*x + 9*n*x^2)*v[i - 1])/(n - 1)); 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_A013610 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and…?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=0 and 0<=k<=2n.?
    
    Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) 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
    
    Parent-row provers of polytool polytool ? step › direct check
    
    ↑ Iterated real-rootedness preservers (above)
    
    ↑ Parent-row provers of polytool (above)

Related ?

Same fingerprint ?

A272866 → D → basis A013610

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.