Real-rooted atlas

A038321 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(18 + 50\,x + 32\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 324 + 1152\,x + 1024\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0324115210242500
15832311045529632768125000
21049767464961990656235929610485766250000
3188956816796160597196801061683209437184033554432312500000
4340122243627970561612431360382205952050960793603623878656107374182415625000000
561222003276187381764063327027212039487488021403533312022830435532813529146982434359738368781250000000
611019960576156728328192975198486528346737239654477052719923201095860905574497409858273284947802324992109951162777639062500000000
719835929036831737486458882256887925964893619054706688249650812551168443823666757632526013234675712400771988324352178120883699712351843720888321953125000000000
835704672266246347497291776050779978334208024073471210291207489524376535040159776520032747522367059556040704024046319299461120160308795329740806333186975989760112589990684262497656250000000000
9642684100792321256804463771648111715952335257605958184124547072021184654665056256052726251610806681693735558419211878411902928053233254401058038049176289280626985510622986240222928181554839552360287970189639684882812500000000000
101156831381426176246790694704250882413064570441564161429964189891297280571985675956518912016269814782763204608337448010309162762245142064918996765900857134054655519621120451429567648550092802407624360792267161677822201560962170881152921504606846976244140625000000000000
1120822964865671168481241854673289216513324644984841830433461162043456356352148716275748694917120475892082395823734784112804049160491551948820054053184087387013122673873757878318268416264086297074401804288018779470014179683860489105197582632573992962697836320780021923843689348814741910323212207031250000000000000
1237481336758208102493286882598206832641077981754468167843847665648031773638000643747650148867111911424133249783070830645739523553327548555483886387272194591462714593247232112302697830889367273472133099493725498509361152118310661089332008321024764836596941136215408643399273764182827624038492971590131496140144641180591620717411303424610351562500000000000000
13674664061647745843217991041643939889152022388851823569639833601724770807149068550144091987776381283656007680359774414291242743496704106599826456664516591616024365674618666175220940804331675487762875594833920598947721764743292125184063887756988239284493352965162647029352669454008320305934638776454486163456012551164667751978919526403187597375937010519244803777893186295716170956830517578125000000000000000
14121439531096594251776345427999563645871718446057066608486116229120382103071122255186493440220770663315080774418432094195483014434463751864323070075001951938077838540877970158779731760707010560155940317559463521414021120246424205526065811617218560306661233543548565568094208297368468890713760550879232220272939919047230037688320120491180810418997627453440459014022134929514771251201088033237653166257235558412089258196146291747061761525878906250000000000000000
Technical details
Node id
n-2d67e83eace19da6 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f3c17ca3ebbc301e

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:            324,             1152,              1024;
  1:           5832,            31104,             55296,             32768;
  2:         104976,           746496,           1990656,           2359296,            1048576;
  3:        1889568,         16796160,          59719680,         106168320,           94371840,           33554432;
  4:       34012224,        362797056,        1612431360,        3822059520,         5096079360,         3623878656,         1073741824;
  5:      612220032,       7618738176,       40633270272,      120394874880,       214035333120,       228304355328,       135291469824,         34359738368;
  6:    11019960576,     156728328192,      975198486528,     3467372396544,      7705271992320,     10958609055744,      9740985827328,       4947802324992,       1099511627776;
  7:   198359290368,    3173748645888,    22568879259648,    93619054706688,    249650812551168,    443823666757632,    526013234675712,     400771988324352,     178120883699712,     35184372088832;
  8:  3570467226624,   63474972917760,   507799783342080,  2407347121029120,   7489524376535040,  15977652003274752,  23670595560407040,   24046319299461120,   16030879532974080,   6333186975989760,   1125899906842624;
  9: 64268410079232, 1256804463771648, 11171595233525760, 59581841245470720, 211846546650562560, 527262516108066816, 937355584192118784, 1190292805323325440, 1058038049176289280, 626985510622986240, 222928181554839552, 36028797018963968;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (18+50*x+32*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 324 + 1152*x + 1024*x^2.
%t Mathematica
P[0, x_] := 324 + 1152*x + 1024*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((18 + 50*x + 32*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 324 + 1152*x + 1024*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((18 + 50*x + 32*x^2)*v[i - 1])/(1 + 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

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.