Real-rooted atlas

A026729 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-20 - 32\,x + 20\,n + 12\,n x - 16\,n x^{2} + 20\,n^{2} x + 8\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(100 + 160\,x + 48\,x^{2} - 40\,n + 116\,n x + 200\,n x^{2} + 64\,n x^{3} - 60\,n^{2} x + 8\,n^{2} x^{2} + 32\,n^{2} x^{3} - 16\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(25 + 90\,x + 92\,x^{2} + 24\,x^{3} + 50\,n x + 160\,n x^{2} + 136\,n x^{3} + 32\,n x^{4} + 20\,n^{2} x^{2} + 48\,n^{2} x^{3} + 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2145
32121226
43286032123
556019222480561
681205169287201922484
713232124830403680211244810773
8214362820867214320128645824102445981

All 15 rows

Technical details
Node id
n-db6ca6aa59282404 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-25ca435b56e4f7cb

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:  1;
   2:  1,    4;
   3:  2,   12,    12;
   4:  3,   28,    60,    32;
   5:  5,   60,   192,   224,     80;
   6:  8,  120,   516,   928,    720,    192;
   7: 13,  232,  1248,  3040,   3680,   2112,    448;
   8: 21,  436,  2820,  8672,  14320,  12864,   5824,   1024;
   9: 34,  804,  6072, 22560,  47280,  58176,  41216,  15360,  2304;
  10: 55, 1460, 12612, 54944, 139680, 218880, 212800, 123904, 39168, 5120;
%F Formula
The row polynomials satisfy (20*n-20+(20*n^2+12*n-32)*x+(8*n^3+8*n^2-16*n)*x^2)*P(n,x) = -(40*n-100+(60*n^2-116*n-160)*x+(16*n^3-8*n^2-200*n-48)*x^2-(32*n^2+64*n)*x^3)*P(n-1,x) + (25+(50*n+90)*x+(20*n^2+160*n+92)*x^2+(48*n^2+136*n+24)*x^3+(16*n^2+32*n)*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(40*n - 100 + (60*n^2 - 116*n - 160)*x + (16*n^3 - 8*n^2 - 200*n - 48)*x^2 - (32*n^2 + 64*n)*x^3)*P[n - 1, x] + (25 + (50*n + 90)*x + (20*n^2 + 160*n + 92)*x^2 + (48*n^2 + 136*n + 24)*x^3 + (16*n^2 + 32*n)*x^4)*D[P[n - 1, x], x])/(20*n - 20 + (20*n^2 + 12*n - 32)*x + (8*n^3 + 8*n^2 - 16*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (-(40*n - 100 + (60*n^2 - 116*n - 160)*x + (16*n^3 - 8*n^2 - 200*n - 48)*x^2 - (32*n^2 + 64*n)*x^3)*v[i - 1] + (25 + (50*n + 90)*x + (20*n^2 + 160*n + 92)*x^2 + (48*n^2 + 136*n + 24)*x^3 + (16*n^2 + 32*n)*x^4)*deriv(v[i - 1], x))/(20*n - 20 + (20*n^2 + 12*n - 32)*x + (8*n^3 + 8*n^2 - 16*n)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.