Real-rooted atlas

A249139 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(78 + 32\,x + 2\,x^{2} + 78\,n + 45\,n x + 5\,n x^{2} + 13\,n^{2} x + 4\,n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(78 + 32\,x + 2\,x^{2} + 546\,n + 510\,n x + 145\,n x^{2} + 12\,n x^{3} + 130\,n^{2} x + 74\,n^{2} x^{2} + 10\,n^{2} x^{3} + 7\,n^{3} x^{2} + 2\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1521 - 2223\,x - 1190\,x^{2} - 278\,x^{3} - 24\,x^{4} - 507\,n x - 520\,n x^{2} - 177\,n x^{3} - 20\,n x^{4} - 39\,n^{2} x^{2} - 25\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
27411
31918340
44044993
597136454282
621734813816719
75089504778852028
8115924281413320255345
92683629442691220150614622
106160158961215040646253638931
111420940144343981341626002347105008
123268910015294692418089555109249280037
13753162490022581021279203412549143438749730
14173383614836692406379600115050196561764641996759
Technical details
Node id
n-4bd8a495ad4c6c77 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f6f881c7ee4ae9bf

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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:    1;
  1:    4,    2;
  2:    7,    4;
  3:   19,   18,    3;
  4:   40,   44,    9;
  5:   97,  136,   45,    4;
  6:  217,  348,  138,   16;
  7:  508,  950,  477,   88,   5;
  8: 1159, 2428, 1413,  320,  25;
  9: 2683, 6294, 4269, 1220, 150, 6;
%F Formula
The row polynomials satisfy (78*n+78+(13*n^2+45*n+32)*x+(n^3+4*n^2+5*n+2)*x^2)*P(n,x) = (546*n+78+(130*n^2+510*n+32)*x+(7*n^3+74*n^2+145*n+2)*x^2+(2*n^3+10*n^2+12*n)*x^3)*P(n-1,x) - (1521+(507*n+2223)*x+(39*n^2+520*n+1190)*x^2+(25*n^2+177*n+278)*x^3+(4*n^2+20*n+24)*x^4)*d/dx 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[((546*n + 78 + (130*n^2 + 510*n + 32)*x + (7*n^3 + 74*n^2 + 145*n + 2)*x^2 + (2*n^3 + 10*n^2 + 12*n)*x^3)*P[n - 1, x] - (1521 + (507*n + 2223)*x + (39*n^2 + 520*n + 1190)*x^2 + (25*n^2 + 177*n + 278)*x^3 + (4*n^2 + 20*n + 24)*x^4)*D[P[n - 1, x], x])/(78*n + 78 + (13*n^2 + 45*n + 32)*x + (n^3 + 4*n^2 + 5*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] = ((546*n + 78 + (130*n^2 + 510*n + 32)*x + (7*n^3 + 74*n^2 + 145*n + 2)*x^2 + (2*n^3 + 10*n^2 + 12*n)*x^3)*v[i - 1] - (1521 + (507*n + 2223)*x + (39*n^2 + 520*n + 1190)*x^2 + (25*n^2 + 177*n + 278)*x^3 + (4*n^2 + 20*n + 24)*x^4)*deriv(v[i - 1], x))/(78*n + 78 + (13*n^2 + 45*n + 32)*x + (n^3 + 4*n^2 + 5*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.