Real-rooted atlas

A075497 → θ+1 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(-5\,x^{2} - 14\,x^{3} - 13\,x^{4} - 4\,x^{5} + 2\,n x^{2} + 6\,n x^{3} + 6\,n x^{4} + 2\,n x^{5}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 8\,x^{3} - 12\,x^{4} - 8\,x^{5} - 2\,x^{6}\right)\,P_{n-2}'(x) + \left(2 + 10\,x + 16\,x^{2} + 10\,x^{3} + 2\,x^{4} - 2\,n x - 6\,n x^{2} - 6\,n x^{3} - 2\,n x^{4}\right)\,P_{n-1}(x) + \left(2\,x + 8\,x^{2} + 12\,x^{3} + 8\,x^{4} + 2\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
24201943
3880172104364
416304111614646413541
532115264481379212838438038642
66444163556811059216436811778832803465599
7128171521929608208641734416195945611367282667686128472
8256673281044928586457616543600262120642376608811548400233750587384745
9512266240569420841178624149142304310948544391842048295744640123384202219252361340126558
10102410577923136281628756889613024165123433307712564003841658788417923791988300138397686821894600321969506131
112048421478417493913620113018881118052044836266201600745985708801007051717128928585263250202013424162659575082316939256383011685316
124096168222729884866561414716416095227106560372748003328933985246208156819498291217920404556481379939750032686837810244199910980008258785933137069915405437
1381926720716856532787201003173232648096147123203769225116672112775811584002290619930214432395991413056320338065040002177300493380897123469891522564137902766304020964876137651966814538
1416384268648448326761799687181454868486898916163584377802383472641329683798300163201808040448005450086620698886644558533344005782341556969283514003152060801419065437962803425891895313237452102679392817589088041959
Technical details
Node id
n-94ff839c941ac3e0 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-3ff6292bf8569ca4

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,      4;
  2:   4,     20,      19;
  3:   8,     80,     172,      104;
  4:  16,    304,    1116,     1464,       641;
  5:  32,   1152,    6448,    13792,     12838,      4380;
  6:  64,   4416,   35568,   110592,    164368,    117788,     32803;
  7: 128,  17152,  192960,   820864,   1734416,   1959456,   1136728,    266768;
  8: 256,  67328, 1044928,  5864576,  16543600,  26212064,  23766088,  11548400,   2337505;
  9: 512, 266240, 5694208, 41178624, 149142304, 310948544, 391842048, 295744640, 123384202, 21925236;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = ((2*n-5)*x^2+(6*n-14)*x^3+(6*n-13)*x^4+(2*n-4)*x^5)*P(n-2,x) - (2*x^2+8*x^3+12*x^4+8*x^5+2*x^6)*d/dx P(n-2,x) + (2-(2*n-10)*x-(6*n-16)*x^2-(6*n-10)*x^3-(2*n-2)*x^4)*P(n-1,x) + (2*x+8*x^2+12*x^3+8*x^4+2*x^5)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[(((2*n - 5)*x^2 + (6*n - 14)*x^3 + (6*n - 13)*x^4 + (2*n - 4)*x^5)*P[n - 2, x] - (2*x^2 + 8*x^3 + 12*x^4 + 8*x^5 + 2*x^6)*D[P[n - 2, x], x] + (2 - (2*n - 10)*x - (6*n - 16)*x^2 - (6*n - 10)*x^3 - (2*n - 2)*x^4)*P[n - 1, x] + (2*x + 8*x^2 + 12*x^3 + 8*x^4 + 2*x^5)*D[P[n - 1, x], x])/(1 + 2*x + 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; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = (((2*n - 5)*x^2 + (6*n - 14)*x^3 + (6*n - 13)*x^4 + (2*n - 4)*x^5)*v[i - 2] - (2*x^2 + 8*x^3 + 12*x^4 + 8*x^5 + 2*x^6)*deriv(v[i - 2], x) + (2 - (2*n - 10)*x - (6*n - 16)*x^2 - (6*n - 10)*x^3 - (2*n - 2)*x^4)*v[i - 1] + (2*x + 8*x^2 + 12*x^3 + 8*x^4 + 2*x^5)*deriv(v[i - 1], x))/(1 + 2*x + 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.