Real-rooted atlas

A034870 → D → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + 7\,x - 4\,x^{2} + n - 13\,n x + 12\,n x^{2} + 6\,n^{2} x - 12\,n^{2} x^{2} + 4\,n^{3} x^{2}\right)\,P_{n}(x) = \left(n + n x - n x^{2} - n x^{3} + 6\,n^{2} x + 12\,n^{2} x^{2} + 6\,n^{2} x^{3} + 4\,n^{3} x^{2} + 8\,n^{3} x^{3} + 4\,n^{3} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
12810
244810864224
361205409607502162592
482241512448070006048274451222528
5103603240134403150045360411602304072901000166400
61252859403168099000199584271656253440160380660001597217281105920
71472898286406425025064864811771761537536145945810010004844841572483075827446823936

All 15 rows

Technical details
Node id
n-89cce58696f5dce2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-478b333d0be8428f

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, 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:  0;
  1:  2,    8;
  2:  4,   48,   108,     64;
  3:  6,  120,   540,    960,     750,     216;
  4:  8,  224,  1512,   4480,    7000,    6048,     2744,      512;
  5: 10,  360,  3240,  13440,   31500,   45360,    41160,    23040,     7290,     1000;
  6: 12,  528,  5940,  31680,   99000,  199584,   271656,   253440,   160380,    66000,    15972,     1728;
  7: 14,  728,  9828,  64064,  250250,  648648,  1177176,  1537536,  1459458,  1001000,   484484,   157248,    30758,    2744;
  8: 16,  960, 15120, 116480,  546000, 1729728,  3923920,  6589440,  8339760,  8008000,  5813808,  3144960,  1230320,  329280,   54000,   4096;
  9: 18, 1224, 22032, 195840, 1071000, 4009824, 10915632, 22404096, 35443980, 43758000, 42357744, 32078592, 18823896, 8396640, 2754000, 626688, 88434, 5832;
%F Formula
The row polynomials satisfy (n-1+(6*n^2-13*n+7)*x+(4*n^3-12*n^2+12*n-4)*x^2)*P(n,x) = (n+(6*n^2+n)*x+(4*n^3+12*n^2-n)*x^2+(8*n^3+6*n^2-n)*x^3+4*n^3*x^4)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((n + (6*n^2 + n)*x + (4*n^3 + 12*n^2 - n)*x^2 + (8*n^3 + 6*n^2 - n)*x^3 + 4*n^3*x^4)*P[n - 1, x])/(n - 1 + (6*n^2 - 13*n + 7)*x + (4*n^3 - 12*n^2 + 12*n - 4)*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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((n + (6*n^2 + n)*x + (4*n^3 + 12*n^2 - n)*x^2 + (8*n^3 + 6*n^2 - n)*x^3 + 4*n^3*x^4)*v[i - 1])/(n - 1 + (6*n^2 - 13*n + 7)*x + (4*n^3 - 12*n^2 + 12*n - 4)*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. [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.