Real-rooted atlas

A272866 → D → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-3 + 5\,x + 9\,x^{2} + 4\,x^{3} + 3\,n - 14\,n x - 21\,n x^{2} - 8\,n x^{3} + 9\,n^{2} x + 12\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n}(x) = \left(3\,n + 13\,n x + 18\,n x^{2} + 13\,n x^{3} + 3\,n x^{4} + 9\,n^{2} x + 39\,n^{2} x^{2} + 49\,n^{2} x^{3} + 24\,n^{2} x^{4} + 4\,n^{2} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
1347
26445416120
39120405480225361275
41223212963120360020885886411000
5153802970109602182524660161706080121510084375
6185645670283208145014360415964211328051030141002178144600000
7217849639608162299505591889056679941127450383801001295912822435491964046875

All 15 rows

Technical details
Node id
n-dc3ccd6c0c74356a · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-b8517a80dcb1c5d1

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:  3,    4;
  2:  6,   44,    54,     16;
  3:  9,  120,   405,    480,     225,      36;
  4: 12,  232,  1296,   3120,    3600,    2088,      588,       64;
  5: 15,  380,  2970,  10960,   21825,   24660,    16170,     6080,     1215,      100;
  6: 18,  564,  5670,  28320,   81450,  143604,   159642,   113280,    51030,    14100,     2178,      144;
  7: 21,  784,  9639,  60816,  229950,  559188,   905667,   994112,   745038,   380100,   129591,    28224,    3549,     196;
  8: 24, 1040, 15120, 115360,  541800, 1689408,  3639720,  5536960,  6016680,  4692800,  2622312,  1038240,  283920,   50960,   5400,   256;
  9: 27, 1332, 22356, 200160, 1124550, 4289544, 11600064, 22802112, 33056505, 35628300, 28645056, 17158176, 7601958, 2451960, 558900, 85248, 7803, 324;
%F Formula
The row polynomials satisfy (3*n-3+(9*n^2-14*n+5)*x+(12*n^2-21*n+9)*x^2+(4*n^2-8*n+4)*x^3)*P(n,x) = (3*n+(9*n^2+13*n)*x+(39*n^2+18*n)*x^2+(49*n^2+13*n)*x^3+(24*n^2+3*n)*x^4+4*n^2*x^5)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((3*n + (9*n^2 + 13*n)*x + (39*n^2 + 18*n)*x^2 + (49*n^2 + 13*n)*x^3 + (24*n^2 + 3*n)*x^4 + 4*n^2*x^5)*P[n - 1, x])/(3*n - 3 + (9*n^2 - 14*n + 5)*x + (12*n^2 - 21*n + 9)*x^2 + (4*n^2 - 8*n + 4)*x^3)]];
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] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((3*n + (9*n^2 + 13*n)*x + (39*n^2 + 18*n)*x^2 + (49*n^2 + 13*n)*x^3 + (24*n^2 + 3*n)*x^4 + 4*n^2*x^5)*v[i - 1])/(3*n - 3 + (9*n^2 - 14*n + 5)*x + (12*n^2 - 21*n + 9)*x^2 + (4*n^2 - 8*n + 4)*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. [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.