A200536 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(6\,n + 21\,n x + 18\,n x^{2}\right)\,P_{n-1}(x)\)

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

\(P_{1}(x) = 21 + 36\,x\)

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	21	36																											57
2	252	1314	2268	1296																									5130
3	2268	19764	68607	118584	102060	34992																							346275
4	18144	221616	1156680	3344004	5783400	5983632	3429216	839808																					20776500
5	136080	2138400	14900760	60429240	157180905	271931580	312915960	230947200	99202320	18895680																			1168678125
6	979776	18825696	164112480	856779120	2976375780	7224137766	12500778276	15422024160	13293110880	7624406880	2618941248	408146688																	63108618750
7	6858432	155784384	1630592208	10413549216	45272080980	141482532492	327000714579	565930129968	733407711876	702914572080	484285885776	227133631872	64997360064	8571080448															3313202484375
8	47029248	1232838144	15061116672	113746114944	593905919040	2270917464480	6569182774224	14639178709512	25338276414864	34063761967200	35278011590976	27640305931392	15859355855616	6291173048832	1542794480640	176319369216													170393270625000
9	317447424	9432723456	131740680960	1148570129664	7001104531680	31663079050176	110019003684912	299997477985968	649909913990589	1124990542447380	1555983052115184	1709806268709504	1474432614371808	976858895279232	480194782099200	165034929586176	35407133330688	3570467226624											8626159325390625
10	2116316160	70291929600	1104717035520	10919737889280	76108813522560	397417702000320	1612287293037120	5200354853557920	13533053986641420	28664451169148970	49621197951018540	70204790523031920	80844691408004160	75111945678060480	55483325057946240	31841955685140480	13690758221199360	4150668150950400	791453568568320	71409344532480									431307966269531250
11	13967686656	512813638656	8956779068160	98981013795840	776437511794560	4597281235780992	21328399557999072	79435049002246080	241369641831156300	604765950908246820	1257603197419563687	2177157423269688552	3137805343805031900	3753306065356127280	3702000780424124640	2979038240786082816	1924478016733996416	974121647271759360	372181040619252480	100937108496660480	17320336516353024	1413905021743104							21349744330341796875
12	91424858112	3676585365504	70648559106048	863137187675136	7525580054952960	49822400084396544	260170012921851648	1098826633064198016	3818470987721551392	11047842617393664720	26825844770783322264	54936435056484448908	95109813278231778936	139202816979160175472	171831194447469812640	178009914556400078592	153537474768595593984	108961588984575241728	62542085604692069376	28315215441682836480	9734029122190399488	2388085581724102656	372499704819228672	27763953154228224					1048078358034960937500
13	594261577728	25977720397824	544640735987712	7289339854178304	69930288279371520	511883721958422528	2971313496983639808	14032777710535144704	54891682574376696048	180108311328475976160	500166594816367719240	1182807727441645961448	2391121656873802559889	4139827046045760865068	6138408209109967463400	7780679049390162170112	8398427433879634495344	7672421213235090366912	5879380463817199282944	3731632333076900229120	1927013009884017479424	789118419923853566976	246563762948261111808	55222502823759937536	7895374178233651200	541397086507450368			51093819954204345703125
14	3839844040704	181295493636096	4118232733655040	59921177667047424	627266122928040960	5030357772453511680	32131275434523353088	167793399517820364288	729639152644988634048	2677193293193352853920	8369118577222288677792	22445547879683562785424	51895513302997537918836	103754060484925895965998	179638315279606862026740	269346574556202753425088	349220493358639136646048	390334782147590846101536	374304885306879169266624	305803470621227613914880	210813298125907719610368	121015316931914130485760	56793803848926098992128	21229713720368900038656	6079438117239911424000	1252522159634986426368	165292695103697731584	10494774292298268672	2476085120857595214843750

Technical details

Node id: n-d5de6dc9110cf4eb · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-9a572e9b19125e69

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:        21,         36;
  2:       252,       1314,         2268,          1296;
  3:      2268,      19764,        68607,        118584,        102060,          34992;
  4:     18144,     221616,      1156680,       3344004,       5783400,        5983632,         3429216,          839808;
  5:    136080,    2138400,     14900760,      60429240,     157180905,      271931580,       312915960,       230947200,        99202320,         18895680;
  6:    979776,   18825696,    164112480,     856779120,    2976375780,     7224137766,     12500778276,     15422024160,     13293110880,       7624406880,       2618941248,        408146688;
  7:   6858432,  155784384,   1630592208,   10413549216,   45272080980,   141482532492,    327000714579,    565930129968,    733407711876,     702914572080,     484285885776,     227133631872,      64997360064,      8571080448;
  8:  47029248, 1232838144,  15061116672,  113746114944,  593905919040,  2270917464480,   6569182774224,  14639178709512,  25338276414864,   34063761967200,   35278011590976,   27640305931392,   15859355855616,   6291173048832,   1542794480640,    176319369216;
  9: 317447424, 9432723456, 131740680960, 1148570129664, 7001104531680, 31663079050176, 110019003684912, 299997477985968, 649909913990589, 1124990542447380, 1555983052115184, 1709806268709504, 1474432614371808, 976858895279232, 480194782099200, 165034929586176, 35407133330688, 3570467226624;

%F Formula

The row polynomials satisfy (n-1)*P(n,x) = (6*n+21*n*x+18*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 21 + 36*x.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 21 + 36*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n + 21*n*x + 18*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] = 21 + 36*x; for(n = 2, N, my(i = n + 1); v[i] = ((6*n + 21*n*x + 18*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 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

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.