Real-rooted atlas

A013622 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 5\,n x\right)\,P_{n}(x) = \left(9 + 30\,x + 25\,x^{2} + 15\,n x + 25\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131013
296075144
3272706755001472
481108040506000312514336
5243405020250450004687518750135168
672914580911252700004218753375001093751245184
7218751030382725141750029531253543750229687562500011272192

All 15 rows

Technical details
Node id
n-cd2143df0e79ad41 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e422756c786abb1e

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:     3,     10;
  2:     9,     60,      75;
  3:    27,    270,     675,      500;
  4:    81,   1080,    4050,     6000,     3125;
  5:   243,   4050,   20250,    45000,    46875,     18750;
  6:   729,  14580,   91125,   270000,   421875,    337500,    109375;
  7:  2187,  51030,  382725,  1417500,  2953125,   3543750,   2296875,    625000;
  8:  6561, 174960, 1530900,  6804000, 17718750,  28350000,  27562500,  15000000,  3515625;
  9: 19683, 590490, 5904900, 30618000, 95681250, 191362500, 248062500, 202500000, 94921875, 19531250;
%F Formula
The row polynomials satisfy (3+5*n*x)*P(n,x) = (9+(15*n+30)*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[((9 + (15*n + 30)*x + (25*n + 25)*x^2)*P[n - 1, x])/(3 + 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] = ((9 + (15*n + 30)*x + (25*n + 25)*x^2)*v[i - 1])/(3 + 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.