Real-rooted atlas

A038337 → θ+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(12 + 22\,x + 11\,n x\right)\,P_{n}(x) = \left(144 + 528\,x + 363\,x^{2} + 132\,n x + 121\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 528\,x + 363\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01445283631035
11728950413068532429624
22073615206431363225555273205815189
3248832228096062726407666560439230096630621827598
42985984328458241129075201839974401581228006957403212400927572834527
5358318084598415361896846336386394624044274384002922109344104167786815589736814803588900
6429981696630639820830349541376741877678081062585216009350749900850000537664149661473281929229929377935624617
75159780352851363758084682500669441335379820544229518406656025247024732161800019355904808171955712208356832332235794769109553940204282
861917364224113515167744070237510041602289222549504045903681331200605928593571845400058067712032326878228480125014099399202829537229200285311670611239553303974579
9743008370688149840021422081030150147276803777217206681608656122765312001333042905858048142561532987596810667869815398405500620373564801867494571272003766114052065234237400473325965417614765456
1089161004482561961542098616321483416212078592604354753069056015581020977561600274225969205084163421476791702323230723465068347392198022333448332808963973942105600271160211748694449301856681580840799568897373147685512476658685
11106993205379072255000472820121621037539007660032942793414787727362700710302777344005347406399499141127625005421508034567988100917770321926178296803587983363495949837421184001410033101093210883845544821163302463647327479901884833179700150343637827655715373286
12128391846454886432953907256754176294525546107240448143990266985761996845371933086659379209981825279065063424160125113851668725761917144220264877260817299231050046353408117463914537351782405922139024591485696215350509985144934453463755083117579281197418962525712569624750374861589214436062754726791
131540702157458636842369309330112512040780460230233292802159854004786429952074244981414533529600179672855023171141632320250227703337451520431357449559597383680444837369858334801920352391743612055347200213197004885293485056969077294933152204803207825304987054752073077677066273140801025324550674750700668359707106504162179453229013284964092
141848842588950364165423271594254401536559274883157485158403189938222453804236801187919702632536473600313610801494989629030461488043719040790691849202292257271410851840106760968766000352460809665602110502089523200682230415632939152179237212568125433044664321539756146393786280960467697133224148101120984311568647760672001283250637644487987278114540768072673753060411857413857875793
Technical details
Node id
n-e3e850d58b0f9c5f · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-4df2d4bb71a9e1dc

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          144,            528,             363;
  1:         1728,           9504,           13068,            5324;
  2:        20736,         152064,          313632,          255552,           73205;
  3:       248832,        2280960,         6272640,         7666560,         4392300,           966306;
  4:      2985984,       32845824,       112907520,       183997440,       158122800,         69574032,         12400927;
  5:     35831808,      459841536,      1896846336,      3863946240,      4427438400,       2922109344,       1041677868,        155897368;
  6:    429981696,     6306398208,     30349541376,     74187767808,    106258521600,      93507499008,      50000537664,      14966147328,      1929229929;
  7:   5159780352,    85136375808,    468250066944,   1335379820544,   2295184066560,    2524702473216,    1800019355904,     808171955712,    208356832332,     23579476910;
  8:  61917364224,  1135151677440,   7023751004160,  22892225495040,  45903681331200,   60592859357184,   54000580677120,   32326878228480,  12501409939920,   2829537229200,   285311670611;
  9: 743008370688, 14984002142208, 103015014727680, 377721720668160, 865612276531200, 1333042905858048, 1425615329875968, 1066786981539840, 550062037356480, 186749457127200, 37661140520652, 3423740047332;
%F Formula
The row polynomials satisfy (12+(11*n+22)*x)*P(n,x) = (144+(132*n+528)*x+(121*n+363)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 528*x + 363*x^2.
%t Mathematica
P[0, x_] := 144 + 528*x + 363*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((144 + (132*n + 528)*x + (121*n + 363)*x^2)*P[n - 1, x])/(12 + (11*n + 22)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 528*x + 363*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((144 + (132*n + 528)*x + (121*n + 363)*x^2)*v[i - 1])/(12 + (11*n + 22)*x)); 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

Proof chains ? JSON

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.