Real-rooted atlas

A038320 → x ↦ x+1 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(23 + 6\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 529 + 276\,x + 36\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
052927636841
1121679522248421624389
2279841292008114264198721296707281
36436343839523043801201142640149040777620511149
41480358892317083481511141405256144010283760107308846656594823321
5340482544762175073384865875308211559796055189512086383584751161627993617249876309
67831098528116343162145614922017611277854004928253871755205298193152691068672515082241679616500246412961
7180115266146342287932051744412653779312268596317001610510290665282741814956164768373836853311011843476805121007769614507145975869
841426511213649108069159687780126863796155220882530755862404028944755024012612348798336274181495616040871775744039983258880231787008060466176420707233300201
995280975791392727341497401008343566282269696740279100351541484014561757471729605318207076631681387358367816962585139815808033719214988802932105651201529794252836279705612200509765705829
10219146244320203216860230256980274498429390643630024855907744727217605023806327746712020968930759290624638184849195801614269971783260162326625834227202697537199104021111160688641001319874562176782336353814783205469041
11504036361936467383170934070569758503826754898002223070162559164156734380624166902010221807432078371378712848707227259609987077811271112026053246464139132224886786562016409056330240210407901530112149697321246726508579184641306069401610260628712958602189
12115928363245387498094233905440266326017271792309639298571596749137144062245964485374244729142199310428039537717263648576109719930197988190083271153198449337344746676273559087872129856743227667456169378360731740161606751248048128104788124872704420554347315278364164096297558232675799463481
132666352354643912456071043355269208487482810190525744811984670774021537692891789571478801685558574140053420080967364051245595875872420593065758954728640141068481683127672960368004734825550451207466762735590878720116871068904900710413858229514415104012050634360360960725456249118720270356366131204701849845768629188747598184440949
14613261041568099864896125596982604581559578272500810529220073991748806096823833983509464768051690462940294971549120323627246234890256655361547782481992953401395257681334732656648499201692821780197532075520392538383813920481280716809222616724357121019965692260951654411086583611532083208898929989189632049745571368140801730280743239682821109907456250246473680347348787521
Technical details
Node id
n-93393a81114f8e3d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-a0b02c0a33159b1d

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:             529,              276,               36;
  1:           12167,             9522,             2484,              216;
  2:          279841,           292008,           114264,            19872,             1296;
  3:         6436343,          8395230,          4380120,          1142640,           149040,            7776;
  4:       148035889,        231708348,        151114140,         52561440,         10283760,         1073088,           46656;
  5:      3404825447,       6217507338,       4865875308,       2115597960,        551895120,        86383584,         7511616,         279936;
  6:     78310985281,     163431621456,     149220176112,      77854004928,      25387175520,      5298193152,       691068672,       51508224,       1679616;
  7:   1801152661463,    4228793205174,    4412653779312,    2685963170016,    1051029066528,    274181495616,     47683738368,     5331101184,     347680512,     10077696;
  8:  41426511213649,  108069159687780,  126863796155220,   88253075586240,   40289447550240,  12612348798336,   2741814956160,   408717757440,   39983258880,   2317870080,    60466176;
  9: 952809757913927, 2734149740100834, 3566282269696740, 2791003515414840, 1456175747172960, 531820707663168, 138735836781696, 25851398158080, 3371921498880, 293210565120, 15297942528, 362797056;
%F Formula
The row polynomials satisfy P(n,x) = (23+6*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 276*x + 36*x^2.
%t Mathematica
P[0, x_] := 529 + 276*x + 36*x^2;
P[n_, x_] := P[n, x] = Expand[(23 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 529 + 276*x + 36*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 6*x)*v[i - 1]); 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.