Real-rooted atlas

A013620 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 60\,x + 180\,n x + 144\,n^{2} x^{2}\right)\,P_{n}(x) = \left(125 + 900\,x + 2160\,x^{2} + 1728\,x^{3} + 900\,n x + 3600\,n x^{2} + 3456\,n x^{3} + 720\,n^{2} x^{2} + 1728\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
154853
22548012961801
31253600194402764850813
4625240001944005529605184001290385
531251500001620000691200012960000895795230603077
6156259000001215000069120000194400000268738560146313216691637401
778125525000085050000604800000226800000047029248005120962560229323571215080301197

All 15 rows

Technical details
Node id
n-76e9f95e7dfa22b6 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-6457d3d804ae805d

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:       5,        48;
  2:      25,       480,       1296;
  3:     125,      3600,      19440,       27648;
  4:     625,     24000,     194400,      552960,       518400;
  5:    3125,    150000,    1620000,     6912000,     12960000,      8957952;
  6:   15625,    900000,   12150000,    69120000,    194400000,    268738560,     146313216;
  7:   78125,   5250000,   85050000,   604800000,   2268000000,   4702924800,    5120962560,    2293235712;
  8:  390625,  30000000,  567000000,  4838400000,  22680000000,  62705664000,  102419251200,   91729428480,   34828517376;
  9: 1953125, 168750000, 3645000000, 36288000000, 204120000000, 705438720000, 1536288768000, 2063912140800, 1567283281920, 515978035200;
%F Formula
The row polynomials satisfy (25+(180*n-60)*x+144*n^2*x^2)*P(n,x) = (125+(900*n+900)*x+(720*n^2+3600*n+2160)*x^2+(1728*n^2+3456*n+1728)*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[((125 + (900*n + 900)*x + (720*n^2 + 3600*n + 2160)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*P[n - 1, x])/(25 + (180*n - 60)*x + 144*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] = ((125 + (900*n + 900)*x + (720*n^2 + 3600*n + 2160)*x^2 + (1728*n^2 + 3456*n + 1728)*x^3)*v[i - 1])/(25 + (180*n - 60)*x + 144*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.