Real-rooted atlas

A038335 → x ↦ x+1 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(7 + 3\,n x\right)\,P_{n}(x) = \left(147 + 126\,x + 27\,x^{2} + 63\,n x + 27\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1211839
24417562431440
392612381415309291651300
4194481666792642978244944328051782000
54084101175032902250423012859560344452535429460750000
6857661214410829087088832455401015202170050754464104437200872041200000
718010885411080653124620841167403198487308601063324867532811167345468527893826375267797000000

All 15 rows

Technical details
Node id
n-26b9fec6ac41e789 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5520686a41bcab10

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:           21,            18;
  2:          441,           756,            243;
  3:         9261,         23814,          15309,           2916;
  4:       194481,        666792,         642978,         244944,          32805;
  5:      4084101,      17503290,       22504230,       12859560,        3444525,        354294;
  6:     85766121,     441082908,      708883245,      540101520,      217005075,      44641044,       3720087;
  7:   1801088541,   10806531246,    20841167403,    19848730860,    10633248675,    3281116734,     546852789,     38263752;
  8:  37822859361,  259356749904,   583552687284,   666917356896,   446596444350,  183742537104,   45935634276,   6428310336,   387420489;
  9: 794280046581, 6127303216482, 15755922556668, 21007896742224, 16881345596430, 8681834878164, 2893944959388, 607475326752, 73222472421, 3874204890;
%F Formula
The row polynomials satisfy (7+3*n*x)*P(n,x) = (147+(63*n+126)*x+(27*n+27)*x^2)*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[((147 + (63*n + 126)*x + (27*n + 27)*x^2)*P[n - 1, x])/(7 + 3*n*x)]];
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] = ((147 + (63*n + 126)*x + (27*n + 27)*x^2)*v[i - 1])/(7 + 3*n*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. [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.