Real-rooted atlas

A038305 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(13 + 35\,x + 22\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 169 + 572\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01695724841225
1219711154188761064842875
2285611933364907765536962342561500625
33712933141710106334801799512015226640515363252521875
44826809490106762073528604678731205938389604019832961133799041838265625
562748517743328586377382205210644113480180131151201829023996810317571264249435788864339296875
68157307211104373899265412915568221397560384468340993120634061652224536513705728259413220352548758735362251875390625
71060449937316151468275810933301602084317252427488109591792390081854630332755220924034523392151756733905926420477203712120726921779278815638671875
813785849184923329898620601776661510338080177545081920237448883511840482203886516352680031122010240657612513592320417331018241280156944998312960265599227914242758547353515625
9179216039403733361755027458282291773309260143317361833932048507414774561601149252596197305619448890089492864235096473609254401989277853616768011221567379376640379806895917363258431830141132896549157373046875
10232980851224814731303440257684403751663624456248416760512148809458945881039512025611915000968524850567114232681446473350099766087372877581836291053952058352150372758528029624937881554329691153655020167168128550026310492163379220508056640625
1130287510659225366632523450295666765763919568482441982432526553147217761798376618639205410517043954600864122083461504616634882066027810078127667226222660666376235776246537835324904780801668871500660893900877024838492041256962172495444647317504282810057883082752118272717781982421875
1239373763856992899328553283041392410261408611345531646946184290756975264323264730454459233441094126779999708174722777398749230028443525371672306203131934727954207068800791518728973977205826534021127593365328007067248644672840201850502922241976970854629058928645147143053472106086462218212734278205444139545122369384765625
13511858930140907571299334207280765370153921129170182974601128754947248008480405730602039874504592802133547220999430940704601769728999839496096013093451246370134090880221581482630879192307202916542591893623556864029614124779227562269696227800959840212017459201285031055508888303616050184644771353034342401213255148318425006080136880068015412051968144884079282928466796875
1466541660918317984118017434340959946464228682820481414705120180600791559681356864099330435357824746275204034343836071651233331212516810363196661518796830260420658277643232256057611185484028589999872086662979873410528546816010266229923465554920161289476519929352819926302726682161488646219178803203479468704147143714406401261785354251162006323202847105414720570680934430113614963390651432965070942774902496337890625
Technical details
Node id
n-41bf0d237dcc11ee · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-f0e992bc07a77fb3

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:           169,            572,             484;
  1:          2197,          11154,           18876,            10648;
  2:         28561,         193336,          490776,           553696,           234256;
  3:        371293,        3141710,        10633480,         17995120,         15226640,           5153632;
  4:       4826809,       49010676,       207352860,        467873120,        593838960,         401983296,         113379904;
  5:      62748517,      743328586,      3773822052,      10644113480,      18013115120,       18290239968,       10317571264,        2494357888;
  6:     815730721,    11043738992,     65412915568,     221397560384,     468340993120,      634061652224,      536513705728,      259413220352,       54875873536;
  7:   10604499373,   161514682758,   1093330160208,    4317252427488,   10959179239008,    18546303327552,    20924034523392,    15175673390592,     6420477203712,     1207269217792;
  8:  137858491849,  2332989862060,  17766615103380,   80177545081920,  237448883511840,   482203886516352,   680031122010240,   657612513592320,   417331018241280,   156944998312960,   26559922791424;
  9: 1792160394037, 33361755027458, 282291773309260, 1433173618339320, 4850741477456160, 11492525961973056, 19448890089492864, 23509647360925440, 19892778536167680, 11221567379376640, 3798068959173632, 584318301411328;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (13+35*x+22*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 572*x + 484*x^2.
%t Mathematica
P[0, x_] := 169 + 572*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((13 + 35*x + 22*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] = 169 + 572*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((13 + 35*x + 22*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

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.