Real-rooted atlas

A113953 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(36\,n + 92\,n x + 40\,n x^{2} + 36\,n^{2} x + 56\,n^{2} x^{2} + 16\,n^{3} x^{2}\right)\,P_{n}(x) = \left(180 + 568\,x + 496\,x^{2} + 160\,x^{3} - 144\,n - 80\,n x + 336\,n x^{2} + 224\,n x^{3} - 216\,n^{2} x - 144\,n^{2} x^{2} + 64\,n^{2} x^{3} - 64\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(81 + 432\,x + 776\,x^{2} + 512\,x^{3} + 80\,x^{4} + 162\,n x + 612\,n x^{2} + 632\,n x^{3} + 112\,n x^{4} + 72\,n^{2} x^{2} + 160\,n^{2} x^{3} + 32\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1145
23161231
35448432165
41112033632080867
5212921128172810401924401
643704339671687120307244821951
7851628951625696357602553685121024107757

All 15 rows

Technical details
Node id
n-e54258729337a0aa · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-91bc824524cfca9d

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:   3,   16,    12;
  3:   5,   44,    84,     32;
  4:  11,  120,   336,    320,     80;
  5:  21,  292,  1128,   1728,   1040,    192;
  6:  43,  704,  3396,   7168,   7120,   3072,    448;
  7:  85, 1628,  9516,  25696,  35760,  25536,   8512,   1024;
  8: 171, 3720, 25416,  83520, 150080, 151680,  83328,  22528,  2304;
  9: 341, 8340, 65376, 253440, 558880, 734592, 573440, 253952, 57600, 5120;
%F Formula
The row polynomials satisfy (36*n+(36*n^2+92*n)*x+(16*n^3+56*n^2+40*n)*x^2)*P(n,x) = -(144*n-180+(216*n^2+80*n-568)*x+(64*n^3+144*n^2-336*n-496)*x^2-(64*n^2+224*n+160)*x^3)*P(n-1,x) + (81+(162*n+432)*x+(72*n^2+612*n+776)*x^2+(160*n^2+632*n+512)*x^3+(32*n^2+112*n+80)*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[(-(144*n - 180 + (216*n^2 + 80*n - 568)*x + (64*n^3 + 144*n^2 - 336*n - 496)*x^2 - (64*n^2 + 224*n + 160)*x^3)*P[n - 1, x] + (81 + (162*n + 432)*x + (72*n^2 + 612*n + 776)*x^2 + (160*n^2 + 632*n + 512)*x^3 + (32*n^2 + 112*n + 80)*x^4)*D[P[n - 1, x], x])/(36*n + (36*n^2 + 92*n)*x + (16*n^3 + 56*n^2 + 40*n)*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] = (-(144*n - 180 + (216*n^2 + 80*n - 568)*x + (64*n^3 + 144*n^2 - 336*n - 496)*x^2 - (64*n^2 + 224*n + 160)*x^3)*v[i - 1] + (81 + (162*n + 432)*x + (72*n^2 + 612*n + 776)*x^2 + (160*n^2 + 632*n + 512)*x^3 + (32*n^2 + 112*n + 80)*x^4)*deriv(v[i - 1], x))/(36*n + (36*n^2 + 92*n)*x + (16*n^3 + 56*n^2 + 40*n)*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.