Real-rooted atlas

A030528 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(40 + 128\,x + 40\,n + 208\,n x + 64\,n x^{2} - 256\,n x^{3} + 80\,n^{2} x + 128\,n^{2} x^{2} - 256\,n^{2} x^{3} + 64\,n^{3} x^{2}\right)\,P_{n}(x) = \left(240 + 1088\,x + 1408\,x^{2} - 1024\,x^{3} - 2048\,x^{4} - 80\,n + 624\,n x + 2304\,n x^{2} + 768\,n x^{3} - 2048\,n x^{4} - 240\,n^{2} x + 256\,n^{2} x^{2} + 1280\,n^{2} x^{3} - 128\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(25 + 180\,x + 288\,x^{2} - 512\,x^{3} - 1792\,x^{4} - 1024\,x^{5} + 100\,n x + 560\,n x^{2} + 384\,n x^{3} - 1280\,n x^{4} - 1024\,n x^{5} + 80\,n^{2} x^{2} + 384\,n^{2} x^{3} + 256\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
12810
23324883
3580288256629
481761008204812804520
5133602832921612800614431365
621704705631744704007372828672212325
734133616272936962880004792324014081310721411050

All 15 rows

Technical details
Node id
n-99236eea6c591277 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-eb0da916da09bf7a

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:  2,    8;
  2:  3,   32,    48;
  3:  5,   80,   288,    256;
  4:  8,  176,  1008,   2048,    1280;
  5: 13,  360,  2832,   9216,   12800,    6144;
  6: 21,  704,  7056,  31744,   70400,   73728,    28672;
  7: 34, 1336, 16272,  93696,  288000,  479232,   401408,   131072;
  8: 55, 2480, 35568, 249856,  985600, 2273280,  3010560,  2097152,   589824;
  9: 89, 4528, 74736, 620032, 2991360, 8865792, 16257024, 17825792, 10616832, 2621440;
%F Formula
The row polynomials satisfy (40*n+40+(80*n^2+208*n+128)*x+(64*n^3+128*n^2+64*n)*x^2-(256*n^2+256*n)*x^3)*P(n,x) = -(80*n-240+(240*n^2-624*n-1088)*x+(128*n^3-256*n^2-2304*n-1408)*x^2-(1280*n^2+768*n-1024)*x^3+(2048*n+2048)*x^4)*P(n-1,x) + (25+(100*n+180)*x+(80*n^2+560*n+288)*x^2+(384*n^2+384*n-512)*x^3+(256*n^2-1280*n-1792)*x^4-(1024*n+1024)*x^5)*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[(-(80*n - 240 + (240*n^2 - 624*n - 1088)*x + (128*n^3 - 256*n^2 - 2304*n - 1408)*x^2 - (1280*n^2 + 768*n - 1024)*x^3 + (2048*n + 2048)*x^4)*P[n - 1, x] + (25 + (100*n + 180)*x + (80*n^2 + 560*n + 288)*x^2 + (384*n^2 + 384*n - 512)*x^3 + (256*n^2 - 1280*n - 1792)*x^4 - (1024*n + 1024)*x^5)*D[P[n - 1, x], x])/(40*n + 40 + (80*n^2 + 208*n + 128)*x + (64*n^3 + 128*n^2 + 64*n)*x^2 - (256*n^2 + 256*n)*x^3)]];
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] = (-(80*n - 240 + (240*n^2 - 624*n - 1088)*x + (128*n^3 - 256*n^2 - 2304*n - 1408)*x^2 - (1280*n^2 + 768*n - 1024)*x^3 + (2048*n + 2048)*x^4)*v[i - 1] + (25 + (100*n + 180)*x + (80*n^2 + 560*n + 288)*x^2 + (384*n^2 + 384*n - 512)*x^3 + (256*n^2 - 1280*n - 1792)*x^4 - (1024*n + 1024)*x^5)*deriv(v[i - 1], x))/(40*n + 40 + (80*n^2 + 208*n + 128)*x + (64*n^3 + 128*n^2 + 64*n)*x^2 - (256*n^2 + 256*n)*x^3)); 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.