Real-rooted atlas

A013613 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 6\,n x\right)\,P_{n}(x) = \left(1 + 12\,x + 36\,x^{2} + 6\,n x + 36\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111213
2124108133
31363248641225
41486483456648010633
516010808640324004665688837
617216201728097200279936326592722701
7184226830240226800979776228614422394885764801

All 15 rows

Technical details
Node id
n-6bec0e3cadeadef4 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3e287fd9a7722fe2

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: 1,  12;
  2: 1,  24,  108;
  3: 1,  36,  324,   864;
  4: 1,  48,  648,  3456,   6480;
  5: 1,  60, 1080,  8640,  32400,   46656;
  6: 1,  72, 1620, 17280,  97200,  279936,   326592;
  7: 1,  84, 2268, 30240, 226800,  979776,  2286144,  2239488;
  8: 1,  96, 3024, 48384, 453600, 2612736,  9144576, 17915904,  15116544;
  9: 1, 108, 3888, 72576, 816480, 5878656, 27433728, 80621568, 136048896, 100776960;
%F Formula
The row polynomials satisfy (1+6*n*x)*P(n,x) = (1+(6*n+12)*x+(36*n+36)*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[((1 + (6*n + 12)*x + (36*n + 36)*x^2)*P[n - 1, x])/(1 + 6*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] = ((1 + (6*n + 12)*x + (36*n + 36)*x^2)*v[i - 1])/(1 + 6*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

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.