Real-rooted atlas

A038220 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(25 - 30\,x + 90\,n x + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(125 + 450\,x + 540\,x^{2} + 216\,x^{3} + 450\,n x + 900\,n x^{2} + 432\,n x^{3} + 180\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
152429
225240324589
312518004860345610241
462512000486006912032400162745
53125750004050008640008100002799362437061
61562545000030375008640000121500008398080228614434977349
778125262500021262500756000001417500001469664008001504017915904486212969

All 15 rows

Technical details
Node id
n-272e88c4cfcd9298 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-dc876aec23f20348

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,       24;
  2:      25,      240,       324;
  3:     125,     1800,      4860,       3456;
  4:     625,    12000,     48600,      69120,       32400;
  5:    3125,    75000,    405000,     864000,      810000,      279936;
  6:   15625,   450000,   3037500,    8640000,    12150000,     8398080,     2286144;
  7:   78125,  2625000,  21262500,   75600000,   141750000,   146966400,    80015040,    17915904;
  8:  390625, 15000000, 141750000,  604800000,  1417500000,  1959552000,  1600300800,   716636160,  136048896;
  9: 1953125, 84375000, 911250000, 4536000000, 12757500000, 22044960000, 24004512000, 16124313600, 6122200320, 1007769600;
%F Formula
The row polynomials satisfy (25+(90*n-30)*x+36*n^2*x^2)*P(n,x) = (125+(450*n+450)*x+(180*n^2+900*n+540)*x^2+(216*n^2+432*n+216)*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 + (450*n + 450)*x + (180*n^2 + 900*n + 540)*x^2 + (216*n^2 + 432*n + 216)*x^3)*P[n - 1, x])/(25 + (90*n - 30)*x + 36*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 + (450*n + 450)*x + (180*n^2 + 900*n + 540)*x^2 + (216*n^2 + 432*n + 216)*x^3)*v[i - 1])/(25 + (90*n - 30)*x + 36*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

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.