Real-rooted atlas

A143495 → D+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(4 + 11\,x + 9\,x^{2} + 2\,x^{3} - n x - 2\,n x^{2} - n x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2459
316412683
464253329141787
52561393279424557997697
610247221199192697617989473677865
740963610512883623962424601113258429371816627
8163841764297845451883351265086421964619949441904978893475
96553684937745925821369397324817591281399291956853676519851291020100670529
10262144405027726181827945047542120671553088774512933944991758688726052700391312751184865257
11104857619202809146631112629131062170227395930611371553726873921304620378016048292594932928676731084714497935347
1241943049077116581169248540844610411307427383828253138838423607812314429067325231766289444149273177804145500611516369838184325463827
1316777216428690001446093290626060130667972889583392477511088904451004507965718432601305237169531893345553325721418659535823592792056441164167972433132885649
146710886420260094932442082083916432252594070764220698520925077623464415946167918678152035318076169035544026203455601487357136419568413795102073103210495700483405116498533314787249481
152684354569593176553133496452844102826228430850638635747711718934460171441975697341007755204464578841011803415964781008910291354677394913698243638735201051910137358434360562956642510893291871399682472661038437459
Technical details
Node id
n-b1a66fdc3ca86866 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-402c4469f5466095

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      4,       5;
   3:     16,      41,       26;
   4:     64,     253,      329,      141;
   5:    256,    1393,     2794,     2455,       799;
   6:   1024,    7221,    19919,    26976,     17989,      4736;
   7:   4096,   36105,   128836,   239624,    246011,    132584,     29371;
   8:  16384,  176429,   784545,  1883351,   2650864,   2196461,    994944,    190497;
   9:  65536,  849377,  4592582, 13693973,  24817591,  28139929,  19568536,   7651985,  1291020;
  10: 262144, 4050277, 26181827, 94504754, 212067155, 308877451, 293394499, 175868872, 60527003, 9131275;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (4-(n-11)*x-(2*n-9)*x^2-(n-2)*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4 - (n - 11)*x - (2*n - 9)*x^2 - (n - 2)*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^3 + x^4)*D[P[n - 1, x], x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((4 - (n - 11)*x - (2*n - 9)*x^2 - (n - 2)*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^3 + 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, v[i])))};
T(10)

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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.