Real-rooted atlas

A013618 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(1 + 13\,x + 12\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1 + 24\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0124144169
113643217282197
214886469122073628561
3160144017280103680248832371293
4172216034560311040149299229859844826809
51843024604807257605225472209018883583180862748517
619640329676814515201393459283607552286654464429981696815730721
71108518414515226127363135283225082265612899450883869835264515978035210604499373

All 15 rows

Technical details
Node id
n-02a5702f23ec3d86 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-a8581dd4789601b6

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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,  24,  144;
  1: 1,  36,  432,   1728;
  2: 1,  48,  864,   6912,   20736;
  3: 1,  60, 1440,  17280,  103680,    248832;
  4: 1,  72, 2160,  34560,  311040,   1492992,    2985984;
  5: 1,  84, 3024,  60480,  725760,   5225472,   20901888,    35831808;
  6: 1,  96, 4032,  96768, 1451520,  13934592,   83607552,   286654464,   429981696;
  7: 1, 108, 5184, 145152, 2612736,  31352832,  250822656,  1289945088,  3869835264,   5159780352;
  8: 1, 120, 6480, 207360, 4354560,  62705664,  627056640,  4299816960, 19349176320,  51597803520,  61917364224;
  9: 1, 132, 7920, 285120, 6842880, 114960384, 1379524608, 11824496640, 70946979840, 283787919360, 681091006464, 743008370688;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (1+13*x+12*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 24*x + 144*x^2.
%t Mathematica
P[0, x_] := 1 + 24*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 13*x + 12*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1 + 24*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + 13*x + 12*x^2)*v[i - 1])/(1 + x)); 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. [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.