Real-rooted atlas

A103633 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(14 + 44\,x + 32\,x^{2} - 23\,n - 68\,n x - 48\,n x^{2} + 9\,n^{2} + 24\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 48\,x + 96\,x^{2} + 64\,x^{3} - 112\,n - 486\,n x - 712\,n x^{2} - 352\,n x^{3} + 111\,n^{2} + 454\,n^{2} x + 624\,n^{2} x^{2} + 288\,n^{2} x^{3} - 27\,n^{3} - 108\,n^{3} x - 144\,n^{3} x^{2} - 64\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(4 + 28\,x + 72\,x^{2} + 80\,x^{3} + 32\,x^{4} - 30\,n - 178\,n x - 388\,n x^{2} - 368\,n x^{3} - 128\,n x^{4} + 18\,n^{2} + 102\,n^{2} x + 212\,n^{2} x^{2} + 192\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
26814
310322466
42410414464336
5322004564481601296
6725281512211214403846048
788816309661446720384089621600

All 15 rows

Technical details
Node id
n-68ae4e0393bd3f70 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-1788d34a8315ad53

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   0;
  1:   2;
  2:   6,    8;
  3:  10,   32,    24;
  4:  24,  104,   144,    64;
  5:  32,  200,   456,   448,   160;
  6:  72,  528,  1512,  2112,  1440,   384;
  7:  88,  816,  3096,  6144,  6720,  3840,   896;
  8: 192, 1984,  8640, 20544, 28800, 23808, 10752,  2048;
  9: 224, 2752, 14592, 43584, 80160, 92928, 66304, 26624, 4608;
%F Formula
The row polynomials satisfy (9*n^2-23*n+14+(24*n^2-68*n+44)*x+(16*n^2-48*n+32)*x^2)*P(n,x) = -(27*n^3-111*n^2+112*n-8+(108*n^3-454*n^2+486*n-48)*x+(144*n^3-624*n^2+712*n-96)*x^2+(64*n^3-288*n^2+352*n-64)*x^3)*P(n-1,x) + (18*n^2-30*n+4+(102*n^2-178*n+28)*x+(212*n^2-388*n+72)*x^2+(192*n^2-368*n+80)*x^3+(64*n^2-128*n+32)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 111*n^2 + 112*n - 8 + (108*n^3 - 454*n^2 + 486*n - 48)*x + (144*n^3 - 624*n^2 + 712*n - 96)*x^2 + (64*n^3 - 288*n^2 + 352*n - 64)*x^3)*P[n - 1, x] + (18*n^2 - 30*n + 4 + (102*n^2 - 178*n + 28)*x + (212*n^2 - 388*n + 72)*x^2 + (192*n^2 - 368*n + 80)*x^3 + (64*n^2 - 128*n + 32)*x^4)*D[P[n - 1, x], x])/(9*n^2 - 23*n + 14 + (24*n^2 - 68*n + 44)*x + (16*n^2 - 48*n + 32)*x^2)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(27*n^3 - 111*n^2 + 112*n - 8 + (108*n^3 - 454*n^2 + 486*n - 48)*x + (144*n^3 - 624*n^2 + 712*n - 96)*x^2 + (64*n^3 - 288*n^2 + 352*n - 64)*x^3)*v[i - 1] + (18*n^2 - 30*n + 4 + (102*n^2 - 178*n + 28)*x + (212*n^2 - 388*n + 72)*x^2 + (192*n^2 - 368*n + 80)*x^3 + (64*n^2 - 128*n + 32)*x^4)*deriv(v[i - 1], x))/(9*n^2 - 23*n + 14 + (24*n^2 - 68*n + 44)*x + (16*n^2 - 48*n + 32)*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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.