Real-rooted atlas

A274496 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(32 + 36\,x + 24\,n x + 9\,n x^{2} + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(64 - 120\,x - 216\,x^{2} - 54\,x^{3} + 240\,n x + 90\,n x^{2} - 27\,n x^{3} + 162\,n^{2} x^{2} + 54\,n^{2} x^{3} + 27\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-128\,x - 384\,x^{2} - 360\,x^{3} - 108\,x^{4} - 144\,n x^{2} - 324\,n x^{3} - 162\,n x^{4} - 36\,n^{2} x^{3} - 54\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
122
241216
382432
41696108220
532192216440
66457612968642800
71281152259217285600
825630721036813824648034000
9512614420736276481296068000
101024153606912013824012960046656400000
1120483072013824027648025920093312800000
124096737284147201105920155520011197443265924600000
1381921474568294402211840311040022394886531849200000
14163843440642322432774144014515200156764169144576223948852000000
Technical details
Node id
n-2efb8d79b4c84c75 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-5201f89336acf752

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:   2;
  2:   4,   12;
  3:   8,   24;
  4:  16,   96,   108;
  5:  32,  192,   216;
  6:  64,  576,  1296,   864;
  7: 128, 1152,  2592,  1728;
  8: 256, 3072, 10368, 13824,  6480;
  9: 512, 6144, 20736, 27648, 12960;
%F Formula
The row polynomials satisfy (32+(24*n+36)*x+(9*n^2+9*n)*x^2)*P(n,x) = (64+(240*n-120)*x+(162*n^2+90*n-216)*x^2+(27*n^3+54*n^2-27*n-54)*x^3)*P(n-1,x) - (128*x+(144*n+384)*x^2+(36*n^2+324*n+360)*x^3+(54*n^2+162*n+108)*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[((64 + (240*n - 120)*x + (162*n^2 + 90*n - 216)*x^2 + (27*n^3 + 54*n^2 - 27*n - 54)*x^3)*P[n - 1, x] - (128*x + (144*n + 384)*x^2 + (36*n^2 + 324*n + 360)*x^3 + (54*n^2 + 162*n + 108)*x^4)*D[P[n - 1, x], x])/(32 + (24*n + 36)*x + (9*n^2 + 9*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] = ((64 + (240*n - 120)*x + (162*n^2 + 90*n - 216)*x^2 + (27*n^3 + 54*n^2 - 27*n - 54)*x^3)*v[i - 1] - (128*x + (144*n + 384)*x^2 + (36*n^2 + 324*n + 360)*x^3 + (54*n^2 + 162*n + 108)*x^4)*deriv(v[i - 1], x))/(32 + (24*n + 36)*x + (9*n^2 + 9*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

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.