Real-rooted atlas

A038277 → x ↦ x+1 → D

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(18\,n + 11\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 11\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
11111
2396242638
31069213068399327753
4256608470448287496585641073116
557736801411344012937320527076080525538900455
612471148838106288046574352028462104086967540106293661353735834
72618941248960278457614670920880119540836805478955020133930011613641019745801395717
853875362816230466829824422522521344430347012480262989840960964296083521964306836817148710481517989115192
910909760970245333660918784114081080762881394324320435210651088558880520719885100815910885378082778091097762122152921949524394883139
1021819521940480120007370672640293351350533120418297296130560383439188119680234323948295360954653122684802500281987984038198752594202593742460101595786057345590
1143202653442150426401621547980807260445925694720118318378048358401265349320794944092792283524962564725532957289760165018611206944037816765068258051356100709980313842837672150905575229324321
1284834301304586245702750254363852817425070221667328031945962073056768039045064755958272033405222068986521620414302375491763289110050051749760272280708491457605546458876677840677900529371736376611405206521610467289073169428
13165426887543943168121313050865558323240774664318701547528305950138994759680114206814411177945601116688852020406579279615779264417876484170350342421888768159284214467502696043262379238087152079314361936493112881270688183256844879525787110350595513998382072863
143206736589313359872254757406817672478729341104916647990886420930994350266794393631977908035129824768035175698838642807244828661680535190435532817515471438171932825680279244091621358784272552989199949057606662406402665421408111040106711090356811309640498351795653164976701653741580136821795624737106
Technical details
Node id
n-80e5c51bf45dc008 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5e3baf07c55bade9

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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:             0;
  1:            11;
  2:           396,           242;
  3:         10692,         13068,           3993;
  4:        256608,        470448,         287496,          58564;
  5:       5773680,      14113440,       12937320,        5270760,         805255;
  6:     124711488,     381062880,      465743520,      284621040,       86967540,      10629366;
  7:    2618941248,    9602784576,    14670920880,    11954083680,     5478955020,    1339300116,     136410197;
  8:   53875362816,  230466829824,   422522521344,   430347012480,   262989840960,   96429608352,   19643068368,   1714871048;
  9: 1090976097024, 5333660918784, 11408108076288, 13943243204352, 10651088558880, 5207198851008, 1591088537808, 277809109776, 21221529219;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (18*n+11*n*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 11.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 11;
P[n_, x_] := P[n, x] = Expand[Together[((18*n + 11*n*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 11; for(n = 2, N, my(i = n + 1); v[i] = ((18*n + 11*n*x)*v[i - 1])/(n - 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

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.