Real-rooted atlas

A003506 → (θ+1)² → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 26\,x - 66\,x^{2} + 26\,x^{3} - x^{4} + n - 41\,n x + 146\,n x^{2} - 81\,n x^{3} + 5\,n x^{4} + 15\,n^{2} x - 105\,n^{2} x^{2} + 95\,n^{2} x^{3} - 10\,n^{2} x^{4} + 25\,n^{3} x^{2} - 50\,n^{3} x^{3} + 10\,n^{3} x^{4} + 10\,n^{4} x^{3} - 5\,n^{4} x^{4} + n^{5} x^{4}\right)\,P_{n}(x) = \left(n - 10\,n x + 10\,n x^{3} - n x^{4} + 15\,n^{2} x - 15\,n^{2} x^{2} - 25\,n^{2} x^{3} + 5\,n^{2} x^{4} + 25\,n^{3} x^{2} + 15\,n^{3} x^{3} - 10\,n^{3} x^{4} + 10\,n^{4} x^{3} + 10\,n^{4} x^{4} + n^{5} x^{4} + n^{5} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
223234
3396243342
4419297210242192
5532024305120312511000
6648048601536018750777647232
77672850535840656255443216807181888

All 15 rows

Technical details
Node id
n-447f925b1b43db2f · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cc9aba5f97e7a661

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -1)
  • 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: 0;
  1: 1;
  2: 2,   32;
  3: 3,   96,   243;
  4: 4,  192,   972,   1024;
  5: 5,  320,  2430,   5120,   3125;
  6: 6,  480,  4860,  15360,  18750,   7776;
  7: 7,  672,  8505,  35840,  65625,  54432,  16807;
  8: 8,  896, 13608,  71680, 175000, 217728, 134456,  32768;
  9: 9, 1152, 20412, 129024, 393750, 653184, 605052, 294912, 59049;
%F Formula
The row polynomials satisfy (n-1+(15*n^2-41*n+26)*x+(25*n^3-105*n^2+146*n-66)*x^2+(10*n^4-50*n^3+95*n^2-81*n+26)*x^3+(n^5-5*n^4+10*n^3-10*n^2+5*n-1)*x^4)*P(n,x) = (n+(15*n^2-10*n)*x+(25*n^3-15*n^2)*x^2+(10*n^4+15*n^3-25*n^2+10*n)*x^3+(n^5+10*n^4-10*n^3+5*n^2-n)*x^4+n^5*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (15*n^2 - 10*n)*x + (25*n^3 - 15*n^2)*x^2 + (10*n^4 + 15*n^3 - 25*n^2 + 10*n)*x^3 + (n^5 + 10*n^4 - 10*n^3 + 5*n^2 - n)*x^4 + n^5*x^5)*P[n - 1, x])/(n - 1 + (15*n^2 - 41*n + 26)*x + (25*n^3 - 105*n^2 + 146*n - 66)*x^2 + (10*n^4 - 50*n^3 + 95*n^2 - 81*n + 26)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n + (15*n^2 - 10*n)*x + (25*n^3 - 15*n^2)*x^2 + (10*n^4 + 15*n^3 - 25*n^2 + 10*n)*x^3 + (n^5 + 10*n^4 - 10*n^3 + 5*n^2 - n)*x^4 + n^5*x^5)*v[i - 1])/(n - 1 + (15*n^2 - 41*n + 26)*x + (25*n^3 - 105*n^2 + 146*n - 66)*x^2 + (10*n^4 - 50*n^3 + 95*n^2 - 81*n + 26)*x^3 + (n^5 - 5*n^4 + 10*n^3 - 10*n^2 + 5*n - 1)*x^4)); 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.