Real-rooted atlas

A038303 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(121 - 11\,x + 33\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(1331 + 363\,x + 33\,x^{2} + x^{3} + 363\,n x + 55\,n x^{2} + 2\,n x^{3} + 11\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
111415
2121889218
313311452297163096
4146412129665347042543200
516105129282011979019360137536594432
6177156138652241976535425920453752376498087040
71948717149603708304386398198960116462591476377364108988416
82143588816235894724464333721443016962562175026832961660125632811457160192
92357947691771691971663138434042380977984507310650664115765478396278784801910019349176320
1025937424601943179076408681534680537415368320930069525014610546721506558901022208044104511000121255409127424
112853116706111141246682444116718410704556590744584016076916075029464602552364587253830921792017788815665500146411443353857228800
123138428376721136949601893281540683021299482999758723202652691152375555618219552802091958368163353088587030895292820009663061900816943837493870592
1334522712143931163198275589492200288792768922118689654974176421483149766259931675674492163855928636419455991526416789083597104683150046060586135907224167196570630428688384
14379749833583241193327188006013625703728405344991661655169638464649084050640025169942005985752315422662625074280318135808430919812323322424102001773332561697656961860859301842257400363372052480
Technical details
Node id
n-32cec8bc04cf2fe3 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d9fdde5632fc1eed

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          1;
  1:         11,          4;
  2:        121,         88,          9;
  3:       1331,       1452,        297,         16;
  4:      14641,      21296,       6534,        704,        25;
  5:     161051,     292820,     119790,      19360,      1375,       36;
  6:    1771561,    3865224,    1976535,     425920,     45375,     2376,      49;
  7:   19487171,   49603708,   30438639,    8198960,   1164625,    91476,    3773,     64;
  8:  214358881,  623589472,  446433372,  144301696,  25621750,  2683296,  166012,   5632,   81;
  9: 2357947691, 7716919716, 6313843404, 2380977984, 507310650, 66411576, 5478396, 278784, 8019, 100;
%F Formula
The row polynomials satisfy (121+(33*n-11)*x+n^2*x^2)*P(n,x) = (1331+(363*n+363)*x+(11*n^2+55*n+33)*x^2+(n^2+2*n+1)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1331 + (363*n + 363)*x + (11*n^2 + 55*n + 33)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x])/(121 + (33*n - 11)*x + n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((1331 + (363*n + 363)*x + (11*n^2 + 55*n + 33)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1])/(121 + (33*n - 11)*x + n^2*x^2)); 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. [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.