Real-rooted atlas

A075502 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(7 + 8\,x - 6\,x^{2} - 7\,x^{3} - 7\,n x - 14\,n x^{2} - 7\,n x^{3}\right)\,P_{n-1}(x) + \left(7\,x + 21\,x^{2} + 21\,x^{3} + 7\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 49 + 119\,x + 71\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04911971239
1343137217577294201
22401147493106627559884284617
3168071584664966647104514808441253991988631
41176491764735778164115952244170563619264920202624953963799
582354320706224123178503338207261506704639429369458194034267364580101649481905
6576480125612187319873269086998821361137616220211616428351311318595946437394962971986670155586352753
740353607330899577432702892530143670488171356196112433544034136079524900022052312960679507105588740369154538214612038856241983

All 15 rows

Technical details
Node id
n-49aadda7441e1a1e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0890f3b2a4e93b2f

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         49,          119,            71;
  1:        343,         1372,          1757,            729;
  2:       2401,        14749,         31066,          27559,            8842;
  3:      16807,       158466,        496664,         710451,          480844,          125399;
  4:     117649,      1764735,       7781641,       15952244,        17056361,         9264920,         2026249;
  5:     823543,     20706224,     123178503,      338207261,       506704639,       429369458,       194034267,        36458010;
  6:    5764801,    256121873,    1987326908,     6998821361,     13761622021,     16164283513,     11318595946,      4373949629,       719866701;
  7:   40353607,   3308995774,   32702892530,   143670488171,    356196112433,    544034136079,    524900022052,    312960679507,    105588740369,     15453821461;
  8:  282475249,  44106492451,  548549639155,  2954491807134,   8996705854962,  17171469324569,  21413749186578,  17514942881688,   9086499398956,   2718876190205,   358100141148;
  9: 1977326743, 599977428876, 9382657517177, 61296373844069, 225076088988310, 522451377952560, 808783818830470, 850937857119329, 603402428650812, 276832502746702, 74370795456132, 8899677678109;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (7-(7*n-8)*x-(14*n+6)*x^2-(7*n+7)*x^3)*P(n-1,x) + (7*x+21*x^2+21*x^3+7*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 49 + 119*x + 71*x^2.
%t Mathematica
P[0, x_] := 49 + 119*x + 71*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((7 - (7*n - 8)*x - (14*n + 6)*x^2 - (7*n + 7)*x^3)*P[n - 1, x] + (7*x + 21*x^2 + 21*x^3 + 7*x^4)*D[P[n - 1, x], 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] = 49 + 119*x + 71*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((7 - (7*n - 8)*x - (14*n + 6)*x^2 - (7*n + 7)*x^3)*v[i - 1] + (7*x + 21*x^2 + 21*x^3 + 7*x^4)*deriv(v[i - 1], x))/(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

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.