Real-rooted atlas

A038284 → 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

\(\left(1 + x\right)\,P_{n}(x) = \left(14 + 40\,x + 26\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 196 + 728\,x + 676\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01967286761600
1274415288283921757664000
2384162853767949769842564569762560000
3537824499408018549440344489603198832011881376102400000
475295368390054438953824096457088013435094409980355843089157764096000000
510541350413703755527634949504236319865604388797504048903743616302737460488031810176163840000000
61475789056219260088321425190574085293564989441228863301120182573976166416953297786888995627397122088270645766553600000000
72066104678434533463910425653430333441111648647782430967355188224575108024924167120385070489656672452601856263122101365765429503678976262144000000000
8289254654976537187216384044893503083520222329729556480722571621058560161030246978764824921347746713602644714454753280184185470956032076013051505664014116709565337610485760000000000
940495651696648272683132313676817771942912042798472939622401589657566328832041331096724549632767577510598778881018215065080012809454854175742976058530049659361280217397327306199043670344486987776419430400000000000
105669391237529612634643329351681290538568640921679890482820628480333828088929054720991946321389191168214921702967658086434212026186688430083971038753812049920327768278092423168018261375493720719366166178738139463689542895666168217616777216000000000000
117937147732541441916254238285004821352547226604339214540067873354383366750745798343106560225667788116040990725587964277159110246410377647943295490457614454581063875861708814913456653205254144011078567799523903078456112226517069119488173680701124261560322481152873203736576671088640000000000000
121111200682555801628891217746450841634875827136787087362590775730161326489613231461764752488857649145429411937815756813690512479039820103682905741424122737328128472182981419944815820858460750080564596244485428498221766712508416366599879911518247321617020708710177632911364863059631479323688966450997470329715097626843545600000000000000
131555680955578122244333682661967626240563378746055791411204533857527782321356802526006336907293327360103205401765069413089283194452911775958024192076275712383221854863360141654894425983444746240204612625281976086855680227996925314201925353472192464936953547079843840119144960971243430379520510621261305328987340801354709468769240170496016772593422857259253761073741824000000000000000
14217795333780937113664716327752049885184901405993689266257920781218527864030756864047152118288936142110720210163727230686804836352715557452237814597419008189841773042685505437696039663370439275364528947206547604009023234779381760851188521173020521319628886224291755189091770040326672117814389632101253120381263875107978977214464015172746050215489909555203757060926720026072842244360874289942887405977642949672960000000000000000
Technical details
Node id
n-356e0f4b32c06589 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-60d4e40326205ee3

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:           196,            728,             676;
  1:          2744,          15288,           28392,            17576;
  2:         38416,         285376,          794976,           984256,            456976;
  3:        537824,        4994080,        18549440,         34448960,          31988320,          11881376;
  4:       7529536,       83900544,       389538240,        964570880,        1343509440,         998035584,         308915776;
  5:     105413504,     1370375552,      7634949504,      23631986560,       43887975040,       48903743616,       30273746048,         8031810176;
  6:    1475789056,    21926008832,    142519057408,     529356498944,     1228863301120,     1825739761664,     1695329778688,       899562739712,      208827064576;
  7:   20661046784,   345334639104,   2565343033344,   11116486477824,    30967355188224,    57510802492416,    71203850704896,     56672452601856,    26312210136576,     5429503678976;
  8:  289254654976,  5371872163840,  44893503083520,  222329729556480,   722571621058560,  1610302469787648,  2492134774671360,   2644714454753280,  1841854709560320,   760130515056640,   141167095653376;
  9: 4049565169664, 82726831323136, 768177719429120, 4279847293962240, 15896575663288320, 41331096724549632, 76757751059877888, 101821506508001280, 94548541757429760, 58530049659361280, 21739732730619904, 3670344486987776;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (14+40*x+26*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 196 + 728*x + 676*x^2.
%t Mathematica
P[0, x_] := 196 + 728*x + 676*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14 + 40*x + 26*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] = 196 + 728*x + 676*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14 + 40*x + 26*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.