Real-rooted atlas

A038314 → x ↦ x+1 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(34 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1156 + 816\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
011568161442116
1393044161614688172897336
213363361886592998784235008207364477456
3454354248018016056597760199756803525120248832205962976
415448044163271350528288648576013583462403595622405076172829859849474296896
5525233501441297635709441373967221768082160128028525271040604064563271066419235831808435817657216
61785793904896504224161382462286514053124396695109632193971843072054768520396896650330112974625177642998169620047612231936
76071699276646419286574172876827228104714649622423145059123211871076796006441897918103552985833367142414911765217281315743989765159780352922190162669056
8206437775405977672860391319756801157194450372608010891241885859840672694351773696028490584310415368379583620710401690000057958402236764782592017543253196806191736422442420747482776576
97018884363803238427249786353589043248087858271039488050916555816394752035941098223337472017759130886825574462679285482913792158015005419110402788500095631360328058834780160231570942197767430083706881951354384207722496
102386420683693101056101071934838766632961961984617458411110423082171970098954240183299600939021107201035103628832119193642621914128381378561289402444219940864284427009754398720446160015301017604724047220834304303147415240704891610044825689762301673555234816
11811383032455654359043722816266561237647367883610917423797370881020232001078373777408900204706833859215360571894754929745854464269126943496350990336949859800575356436482514334766228884684849300681690762444806960096238695874566699557876819558439409163981291521069932053790724129065876983540801536
122758702310349224820736136312349452549932318723127165663911439623782444148221137573265276928428497440452916986511363024687814961766963609616013053138032883924992645904664391242376806419947055812082485166084693424896960584744968282514524048090726410629965164553699328937938102754738176509287657604382721283918464548864189937030341242876870656
13937958785518736439050244965664158628604677324801226811145072949390868480187629939834686377426944019866699511907969374617601542590785630501151440896907406344488530089082880411764223549417015214080145328549488029534781440398941116241649703321608448164814529052540928135532055848059666432015944947746830548992012986835268911759360654798416919920640154070215745863688737103395697172336050176
14318905987076370389277081618008808681959739629764608476703759228346049023180807851591328466876101558272090062371120649461164933120762881261257266023985315844936290514017603684610867224888859734542539586273280988234136518600836513792031003423890779634086707207659669431839674303774721474588767626889170780162168512893568954662912023549461287626656972801781051694022184140808381419736574984192184884258895036416401906756202069927458308096
Technical details
Node id
n-fe432be4158ab93d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-dde9c6f2b382eac4

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:              1156,                816,                144;
  1:             39304,              41616,              14688,               1728;
  2:           1336336,            1886592,             998784,             235008,              20736;
  3:          45435424,           80180160,           56597760,           19975680,            3525120,             248832;
  4:        1544804416,         3271350528,         2886485760,         1358346240,          359562240,           50761728,           2985984;
  5:       52523350144,       129763570944,       137396722176,        80821601280,        28525271040,         6040645632,         710664192,          35831808;
  6:     1785793904896,      5042241613824,      6228651405312,      4396695109632,      1939718430720,       547685203968,       96650330112,        9746251776,        429981696;
  7:    60716992766464,    192865741728768,    272281047146496,    224231450591232,    118710767960064,     41897918103552,     9858333671424,     1491176521728,     131574398976,      5159780352;
  8:  2064377754059776,   7286039131975680,  11571944503726080,  10891241885859840,   6726943517736960,   2849058431041536,   837958362071040,   169000005795840,   22367647825920,   1754325319680,    61917364224;
  9: 70188843638032384, 272497863535890432, 480878582710394880, 509165558163947520, 359410982233374720, 177591308868255744, 62679285482913792, 15801500541911040, 2788500095631360, 328058834780160, 23157094219776, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (34+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1156 + 816*x + 144*x^2.
%t Mathematica
P[0, x_] := 1156 + 816*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(34 + 12*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1156 + 816*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (34 + 12*x)*v[i - 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

Proof chains ? JSON

Same fingerprint ?

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.