Real-rooted atlas

A008278 → x ↦ x+1 (shift +1) → reverse → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2 - 2\,x + n + 2\,n x\right)\,P_{n}(x) = \left(7\,x^{2} + 8\,x^{3} + 2\,x^{4} - 12\,n x^{2} - 30\,n x^{3} - 21\,n x^{4} - 4\,n x^{5} + 6\,n^{2} x^{2} + 20\,n^{2} x^{3} + 20\,n^{2} x^{4} + 6\,n^{2} x^{5} - n^{3} x^{2} - 4\,n^{3} x^{3} - 5\,n^{3} x^{4} - 2\,n^{3} x^{5}\right)\,P_{n-2}(x) + \left(2\,x^{2} + 5\,x^{3} + 4\,x^{4} + x^{5} - 3\,n x^{2} - 12\,n x^{3} - 17\,n x^{4} - 10\,n x^{5} - 2\,n x^{6} + n^{2} x^{2} + 5\,n^{2} x^{3} + 9\,n^{2} x^{4} + 7\,n^{2} x^{5} + 2\,n^{2} x^{6}\right)\,P_{n-2}'(x) + \left(-2 - 14\,x - 17\,x^{2} - 6\,x^{3} + n + 12\,n x + 23\,n x^{2} + 10\,n x^{3} - 2\,n^{2} x - 6\,n^{2} x^{2} - 4\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x - 6\,x^{2} - 6\,x^{3} - 2\,x^{4} + n x + 4\,n x^{2} + 5\,n x^{3} + 2\,n x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1145
21101526
31206960150
4138228436260963
51726722124272012186807
61138189388161820517256613952448
7126852473388410044515124811279133120437004
8152614514125324500855106578612499487639601903233911237
911040403624552282359135665683810918516104196085374395115975037384873
1012066113259164282410755750386749388321460011024524888309692392924307464270379715078
111411632115359300604816157021479997658239542210045386641110981852764479660298455564505631644080631202
12182149202082149690021375283011598452523855384540831469380811905396083112630732406777759714235347355235937768146225182023
131164082662020784452289459428506155856570245936600846450463563211482314900314003320332011546624273861633815840192469067052672590508550157126907
1413279477645132885416004190627875323530667401530549080134784816550721029768545529155445364931016466873139971201114979208575299110763163060724022207443781756859505297612
Technical details
Node id
n-fc79b5a7d82bc0ba · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-728b95eb80c88fcb

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: 1,    4;
  2: 1,   10,    15;
  3: 1,   20,    69,     60;
  4: 1,   38,   228,    436,     260;
  5: 1,   72,   672,   2124,    2720,    1218;
  6: 1,  138,  1893,   8816,   18205,   17256,     6139;
  7: 1,  268,  5247,  33884,  100445,  151248,   112791,    33120;
  8: 1,  526, 14514, 125324,  500855, 1065786,  1249948,   763960,  190323;
  9: 1, 1040, 40362, 455228, 2359135, 6656838, 10918516, 10419608, 5374395, 1159750;
%F Formula
The row polynomials satisfy (n-2+(2*n-2)*x)*P(n,x) = -((n^3-6*n^2+12*n-7)*x^2+(4*n^3-20*n^2+30*n-8)*x^3+(5*n^3-20*n^2+21*n-2)*x^4+(2*n^3-6*n^2+4*n)*x^5)*P(n-2,x) + ((n^2-3*n+2)*x^2+(5*n^2-12*n+5)*x^3+(9*n^2-17*n+4)*x^4+(7*n^2-10*n+1)*x^5+(2*n^2-2*n)*x^6)*d/dx P(n-2,x) + (n-2-(2*n^2-12*n+14)*x-(6*n^2-23*n+17)*x^2-(4*n^2-10*n+6)*x^3)*P(n-1,x) + ((n-2)*x+(4*n-6)*x^2+(5*n-6)*x^3+(2*n-2)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^3 - 6*n^2 + 12*n - 7)*x^2 + (4*n^3 - 20*n^2 + 30*n - 8)*x^3 + (5*n^3 - 20*n^2 + 21*n - 2)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*P[n - 2, x] + ((n^2 - 3*n + 2)*x^2 + (5*n^2 - 12*n + 5)*x^3 + (9*n^2 - 17*n + 4)*x^4 + (7*n^2 - 10*n + 1)*x^5 + (2*n^2 - 2*n)*x^6)*D[P[n - 2, x], x] + (n - 2 - (2*n^2 - 12*n + 14)*x - (6*n^2 - 23*n + 17)*x^2 - (4*n^2 - 10*n + 6)*x^3)*P[n - 1, x] + ((n - 2)*x + (4*n - 6)*x^2 + (5*n - 6)*x^3 + (2*n - 2)*x^4)*D[P[n - 1, x], x])/(n - 2 + (2*n - 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (-((n^3 - 6*n^2 + 12*n - 7)*x^2 + (4*n^3 - 20*n^2 + 30*n - 8)*x^3 + (5*n^3 - 20*n^2 + 21*n - 2)*x^4 + (2*n^3 - 6*n^2 + 4*n)*x^5)*v[i - 2] + ((n^2 - 3*n + 2)*x^2 + (5*n^2 - 12*n + 5)*x^3 + (9*n^2 - 17*n + 4)*x^4 + (7*n^2 - 10*n + 1)*x^5 + (2*n^2 - 2*n)*x^6)*deriv(v[i - 2], x) + (n - 2 - (2*n^2 - 12*n + 14)*x - (6*n^2 - 23*n + 17)*x^2 - (4*n^2 - 10*n + 6)*x^3)*v[i - 1] + ((n - 2)*x + (4*n - 6)*x^2 + (5*n - 6)*x^3 + (2*n - 2)*x^4)*deriv(v[i - 1], x))/(n - 2 + (2*n - 2)*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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.