Real-rooted atlas

A038333 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(12 + 14\,x + 7\,n x\right)\,P_{n}(x) = \left(144 + 336\,x + 147\,x^{2} + 84\,n x + 49\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 336\,x + 147\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0144336147627
1172860485292137214440
220736967681270086585612005322373
32488321451520254016019756807203001008427037334
42985984209018884572288047416320259308007260624823543151042039
5358318082926264327681443849957427207260624003049462086917761265883443199119908
64299816964013162496122903101441911826022417425497600975827865633205253766324810245188320967040380425
75159780352541776936961896219279363441286840323763907481602634735237121195389135363415397529656033865724035360701392652169362
86191736422472236924928028443289190405899348869120752781496320063233645690883586167406080136615901184033620319432048424328400310722773928719205102331
974300837068895352740904964171682414592097339256340480141953082163200139114020519936946748195205124508324739072014792940550080319600567440041015406154823727920916588582360748896
1089161004482561248254062755846007222677012481557428101447680255515547893760028617741364101122272195668492288129839752485273653254585980288015340827237120029531092431456341682061190417993673361311998496666516557
111069932053790721622730281582592851933397830860824295878382583808442893616349184005580459565999718450637503469256704337583356461711361661543082584985659829226224768001535616806435712266512007728512280701304436281356446145698243464641097277710
121283918464548864209706682542981121192706756963205123710643243885527047440612754666291201041685785653280768106338757285439078481020005550810726446523206312379596820102620011522048064495905870299904149246724327966722357890957264752227882952477264101733460927354920199065084075903
1315407021574586368269622877555261440165144012502597632055659648658282905601217554814399938560018750344141759053824212677514570878156801822950124893241344011963110194611896320603078600345661440023218526113307965446716102594758502401414734574358851202050946572295376018312022966923007596098415908899076829037077631004
1418488425889503641634511728327073464322264832171464196096082205019556848599040194808770303990169600327278734110703484928408340827976086061056388896026643891486720287114644670685511680165415844666238566400742992835625854894082578983396387264921667907259569224857601312605806269040640175795420482460800145845089585448965649598196832171988727640913598577169
Technical details
Node id
n-5e5215f6b5f47c3a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-1b5229a076e1d872

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,           336,            147;
  1:         1728,          6048,           5292,           1372;
  2:        20736,         96768,         127008,          65856,           12005;
  3:       248832,       1451520,        2540160,        1975680,          720300,          100842;
  4:      2985984,      20901888,       45722880,       47416320,        25930800,         7260624,         823543;
  5:     35831808,     292626432,      768144384,      995742720,       726062400,       304946208,       69177612,        6588344;
  6:    429981696,    4013162496,    12290310144,    19118260224,     17425497600,      9758278656,     3320525376,      632481024,       51883209;
  7:   5159780352,   54177693696,   189621927936,   344128684032,    376390748160,    263473523712,   119538913536,    34153975296,     5603386572,     403536070;
  8:  61917364224,  722369249280,  2844328919040,  5899348869120,   7527814963200,   6323364569088,  3586167406080,  1366159011840,   336203194320,   48424328400,   3107227739;
  9: 743008370688, 9535274090496, 41716824145920, 97339256340480, 141953082163200, 139114020519936, 94674819520512, 45083247390720, 14792940550080, 3196005674400, 410154061548, 23727920916;
%F Formula
The row polynomials satisfy (12+(7*n+14)*x)*P(n,x) = (144+(84*n+336)*x+(49*n+147)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 336*x + 147*x^2.
%t Mathematica
P[0, x_] := 144 + 336*x + 147*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((144 + (84*n + 336)*x + (49*n + 147)*x^2)*P[n - 1, x])/(12 + (7*n + 14)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 336*x + 147*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((144 + (84*n + 336)*x + (49*n + 147)*x^2)*v[i - 1])/(12 + (7*n + 14)*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.