Real-rooted atlas

A030528 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + 2\,n + 4\,n x + 2\,n x^{2}\right)\,P_{n}(x) = \left(17 + 76\,x + 122\,x^{2} + 84\,x^{3} + 21\,x^{4} - 9\,n - 44\,n x - 82\,n x^{2} - 68\,n x^{3} - 21\,n x^{4}\right)\,P_{n-1}(x) + \left(5 + 37\,x + 102\,x^{2} + 134\,x^{3} + 85\,x^{4} + 21\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1246
23141532
35357957176
4876264396216960
513155726166618698195248
6213021783552494598478310528672
73457240741590836742501483742211772156672
855106088444169212137022326025326016189244631856064
989193318488102144359042832210127128012335766896611692094677632
101443480375362378569801442743152527832068921285842368290239264152025559040
112336201744635323592516794825839419185022315424143595655727058077120945152432187139657216
1237710956144954115418861580472318607263151564125332344179829639181875996123046938499889889221121763101184
136101922027778824378641449023061582860192478760447844720775507950986876460896871420551114280205200378349599244169662464
149873351452535750381003301348315633437455159733314727778162989655817459059189452475406394329533268243686110583741250613254313522783459328
Technical details
Node id
n-902a8a2088342939 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0ed14db09db6bd99

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:  3,   14,    15;
  3:  5,   35,    79,     57;
  4:  8,   76,   264,    396,    216;
  5: 13,  155,   726,   1666,   1869,    819;
  6: 21,  302,  1783,   5524,   9459,   8478,    3105;
  7: 34,  572,  4074,  15908,  36742,  50148,   37422,   11772;
  8: 55, 1060,  8844,  41692, 121370, 223260,  253260,  161892,  44631;
  9: 89, 1933, 18488, 102144, 359042, 832210, 1271280, 1233576, 689661, 169209;
%F Formula
The row polynomials satisfy (2*n+2+(4*n+4)*x+(2*n+2)*x^2)*P(n,x) = -(9*n-17+(44*n-76)*x+(82*n-122)*x^2+(68*n-84)*x^3+(21*n-21)*x^4)*P(n-1,x) + (5+37*x+102*x^2+134*x^3+85*x^4+21*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(9*n - 17 + (44*n - 76)*x + (82*n - 122)*x^2 + (68*n - 84)*x^3 + (21*n - 21)*x^4)*P[n - 1, x] + (5 + 37*x + 102*x^2 + 134*x^3 + 85*x^4 + 21*x^5)*D[P[n - 1, x], x])/(2*n + 2 + (4*n + 4)*x + (2*n + 2)*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; for(n = 1, N, my(i = n + 1); v[i] = (-(9*n - 17 + (44*n - 76)*x + (82*n - 122)*x^2 + (68*n - 84)*x^3 + (21*n - 21)*x^4)*v[i - 1] + (5 + 37*x + 102*x^2 + 134*x^3 + 85*x^4 + 21*x^5)*deriv(v[i - 1], x))/(2*n + 2 + (4*n + 4)*x + (2*n + 2)*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. [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
  2. [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