A200536 → basis A007318 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 7\,n x + 2\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 7 + 4\,x\)

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	7	4																											11
2	84	146	84	16																									330
3	756	2196	2541	1464	420	48																							7425
4	6048	24624	42840	41284	23800	8208	1568	128																					148500
5	45360	237600	551880	746040	646835	373020	143080	35200	5040	320																			2784375
6	326592	2091744	6078240	10577520	12248460	9909654	5715948	2350560	675360	129120	14784	768																	50118750
7	2286144	17309376	60392304	128562336	186304860	194077548	149520217	86256688	37260972	11903920	2733808	427392	40768	1792															877078125
8	15676416	136982016	557819136	1404273024	2444057280	3115113120	3003741552	2231241992	1287317808	576872800	199145408	52010112	9947392	1315328	107520	4096													15035625000
9	105815808	1048080384	4879284480	14179878144	28811129760	43433578944	50305900176	45724352688	33018844383	19051813620	8783569872	3217302144	924801696	204236928	33465600	3833856	274176	9216											253726171875
10	705438720	7810214400	40915445760	134811578880	313204993920	545154598080	737214125760	792616194720	687550372740	485434997530	280113114820	132102699120	50707849920	15704041920	3866728320	739706880	106014720	10713600	680960	20480									4228769531250
11	4655895552	56979293184	331732558080	1221987824640	3195216097920	6306284274048	9752354621856	12107155769280	12262848236100	10241764482180	7099206858821	4096705792872	1968111445300	784723059120	257998799520	69204765696	14902242432	2514378240	320221440	28948480	1655808	45056							69774697265625
12	30474952704	408509485056	2616613300224	10656014662656	30969465246720	68343484340736	118962054376704	167478529654656	193998424413024	187096184819280	151432678909512	103372594618188	59655297752232	29103850971888	11975211383520	4135272337152	1188921178368	281248906752	53810684928	8120724480	930564096	76099584	3956736	98304					1141767773437500
13	198087192576	2886413377536	20171879110656	89991850051584	287778964112640	702172458104832	1358625284400384	2138816904516864	2788786393048656	3050150067375840	2823455067352920	2225661413857128	1499772415548043	865534994277772	427796222326200	180749633622272	65033427547088	19803860227008	5058569548032	1070221701120	184220644608	25146328064	2619029504	195526656	9318400	212992			18553726318359375
14	1279948013568	20143943737344	152527138283520	739767625519104	2581342069662720	6900353597329920	14691941213773824	25574363590583808	37069509355534656	45338503500370080	47243919328141536	42235258250085264	32550187949993532	21692396602387742	12519303057689820	6257075296308928	2704197402620896	1007522300008224	322048778123072	87703578842880	20153554477056	3856317104640	603270513664	75168178176	7175168000	492756992	21676032	458752	299714040527343750

Technical details

Node id: n-acc13798e1eca257 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-d6c710616865b24c

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:         0;
  1:         7,          4;
  2:        84,        146,         84,          16;
  3:       756,       2196,       2541,        1464,         420,          48;
  4:      6048,      24624,      42840,       41284,       23800,        8208,        1568,         128;
  5:     45360,     237600,     551880,      746040,      646835,      373020,      143080,       35200,        5040,         320;
  6:    326592,    2091744,    6078240,    10577520,    12248460,     9909654,     5715948,     2350560,      675360,      129120,      14784,        768;
  7:   2286144,   17309376,   60392304,   128562336,   186304860,   194077548,   149520217,    86256688,    37260972,    11903920,    2733808,     427392,     40768,      1792;
  8:  15676416,  136982016,  557819136,  1404273024,  2444057280,  3115113120,  3003741552,  2231241992,  1287317808,   576872800,  199145408,   52010112,   9947392,   1315328,   107520,    4096;
  9: 105815808, 1048080384, 4879284480, 14179878144, 28811129760, 43433578944, 50305900176, 45724352688, 33018844383, 19051813620, 8783569872, 3217302144, 924801696, 204236928, 33465600, 3833856, 274176, 9216;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (6*n+7*n*x+2*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 7 + 4*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 7 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 7*n*x + 2*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] = 7 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 7*n*x + 2*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_A007318 of Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle, read by rows of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    Parent-row provers of polytool polytool ? step › direct check
    
    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 of 2*n+1 terms, where row n lists the coefficients in (1+3*x+2*x^2)^n.?
    
    ↑ Parent-row provers of polytool (above)
    
    ↑ Iterated real-rootedness preservers (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.