Real-rooted atlas

A038259 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(6 + 5\,n x\right)\,P_{n}(x) = \left(36 + 60\,x + 25\,x^{2} + 30\,n x + 25\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
161016
23612075231
3216108013505003146
4129686401620012000312541261
57776648001620001800009375018750527076
6466564665601458000216000016875006750001093756603091
7279936326592012247200226800002362500014175000459375062500081491806

All 15 rows

Technical details
Node id
n-67f904f032bfc830 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7d1144c6b7b8d949

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:        6,        10;
  2:       36,       120,        75;
  3:      216,      1080,      1350,        500;
  4:     1296,      8640,     16200,      12000,       3125;
  5:     7776,     64800,    162000,     180000,      93750,      18750;
  6:    46656,    466560,   1458000,    2160000,    1687500,     675000,     109375;
  7:   279936,   3265920,  12247200,   22680000,   23625000,   14175000,    4593750,    625000;
  8:  1679616,  22394880,  97977600,  217728000,  283500000,  226800000,  110250000,  30000000,   3515625;
  9: 10077696, 151165440, 755827200, 1959552000, 3061800000, 3061800000, 1984500000, 810000000, 189843750, 19531250;
%F Formula
The row polynomials satisfy (6+5*n*x)*P(n,x) = (36+(30*n+60)*x+(25*n+25)*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[((36 + (30*n + 60)*x + (25*n + 25)*x^2)*P[n - 1, x])/(6 + 5*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] = ((36 + (30*n + 60)*x + (25*n + 25)*x^2)*v[i - 1])/(6 + 5*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.