Real-rooted atlas

A038337 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(12 + 35\,x + 23\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 552\,x + 529\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01445525291225
117289936190441216742875
2207361589764570565840162798411500625
3248832238464091411201752048016790460643634352521875
42985984343388161645401604204915206044565604634166961480358891838265625
53583180848074342427642746888830321920169247836801946350123212435014676340482544764339296875
6429981696659305267244228395008169542180864406194808320622832039424596880704448326863242912783109852812251875390625
7515978035289006211072682380951552305175925555287738078597121681646506444821487705360128176506151172488457586410348180115266146378815638671875
86191736422411867494809601023571427328052315872952320175476157194240403595161546752644631160803840706024604689920507455184620880216138319375560414265112136492758547353515625
9743008370688156650931486721501238093414408632119037132803308978964234240887909355402854417018262645221376232988119547673602232802812331872014265129078786960546829948020166895280975791392796549157373046875
10891610044825620507031030988821617828545167361381139045941248059561621356216320182655638825730048408438303485313024671005784297299968803809012439473920684726195781774080393717562574520096137204605139605488219146244320203213379220508056640625
1110699320537907226659140340285443065801139132825621545769116683468810324014368410828803561784957101735936910233933481554739217446150391729799168250788411881115863042670432163548918912020473313253875044992107019592008892280643418681411395170076504036361936467383118272717781982421875
121283918464548864344518121320611844292121594785955843290626556002566144173443441389301923846648665253256573747219114912603112649523241870760940151518003270220755326712441651289726520695243675443285987915666275188966459930971524979677158428716923855719428638484678108805326520344115928363245387498094139545122369384765625
1315407021574586368442951870269358080594293759278055424049359398340038492160283816540455221329920119675974558618327449638229825206225299046409420921211534091550720180567656554403421388802691795620857310263296030955649639859068027904269689371862408547212801723015431343165718304076210297924793868309602086710538416974965620266635235464391245607144884079282928466796875
141848842588950364165669783939447783424815031441295618867207290003447144146534404541064647283541278720208888973775042898821127340126439595257416908820097965251272728641536043336237573056821133312073832108457800510078976099058078847549017689292810356071879516488212971528270474070447195447859204877459067186807571814402003242116880295966995205119396520916311915654461326104156809986489615070942774902496337890625
Technical details
Node id
n-81c9d624be1e751d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-4b5617cf7459dd0c

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,            552,             529;
  1:         1728,           9936,           19044,           12167;
  2:        20736,         158976,          457056,          584016,           279841;
  3:       248832,        2384640,         9141120,        17520480,         16790460,          6436343;
  4:      2985984,       34338816,       164540160,       420491520,        604456560,        463416696,         148035889;
  5:     35831808,      480743424,      2764274688,      8830321920,      16924783680,      19463501232,       12435014676,        3404825447;
  6:    429981696,     6593052672,     44228395008,    169542180864,     406194808320,     622832039424,      596880704448,      326863242912,       78310985281;
  7:   5159780352,    89006211072,    682380951552,   3051759255552,    8773807859712,   16816465064448,    21487705360128,    17650615117248,     8457586410348,     1801152661463;
  8:  61917364224,  1186749480960,  10235714273280,  52315872952320,  175476157194240,  403595161546752,   644631160803840,   706024604689920,   507455184620880,   216138319375560,   41426511213649;
  9: 743008370688, 15665093148672, 150123809341440, 863211903713280, 3308978964234240, 8879093554028544, 17018262645221376, 23298811954767360, 22328028123318720, 14265129078786960, 5468299480201668, 952809757913927;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (12+35*x+23*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 552*x + 529*x^2.
%t Mathematica
P[0, x_] := 144 + 552*x + 529*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + 35*x + 23*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] = 144 + 552*x + 529*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + 35*x + 23*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.