Real-rooted atlas

delannoy → x ↦ x+1 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(41 + 273\,x + 750\,x^{2} + 1090\,x^{3} + 885\,x^{4} + 381\,x^{5} + 68\,x^{6} - 57\,n - 384\,n x - 1069\,n x^{2} - 1575\,n x^{3} - 1296\,n x^{4} - 565\,n x^{5} - 102\,n x^{6} + 18\,n^{2} + 120\,n^{2} x + 334\,n^{2} x^{2} + 496\,n^{2} x^{3} + 414\,n^{2} x^{4} + 184\,n^{2} x^{5} + 34\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(52 + 431\,x + 1511\,x^{2} + 2910\,x^{3} + 3330\,x^{4} + 2267\,x^{5} + 851\,x^{6} + 136\,x^{7} - 24\,n - 200\,n x - 707\,n x^{2} - 1376\,n x^{3} - 1594\,n x^{4} - 1100\,n x^{5} - 419\,n x^{6} - 68\,n x^{7}\right)\,P_{n-1}'(x) + \left(8 + 80\,x + 345\,x^{2} + 839\,x^{3} + 1260\,x^{4} + 1198\,x^{5} + 705\,x^{6} + 235\,x^{7} + 34\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
25151131
312548039185
4291743883811391111
5705251564231316984956665
61691521567111211123937263176339991
7408428419184474907018461920301966279239945
89851182061792183798340206401256294490122961223631439671
92378321031918966665821482828219058221490721350099492838796478638025
10574186115579323230166659805861061911813048560109563576016199195081928366751828151
11138602286901709952764777722732536471522246964094873236873537427482620859476444081010295310968905
1233461602298495512424637275824526331956662643376056544267368633921785052555575621120831202970645335982191865813431
138078215752491414088477374935287912862769335813151877279222427548532477302862202156122311845946124720229731146354981281524711194880585
141950254095525398364552378797659742252352894708925643506273510872714435140304706851376059783510096985709537490071919622237334397445514564217967169283511
Technical details
Node id
n-1a5d1ec7417c59f1 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-2a11099686bc538f

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,     3;
  2:    5,    15,     11;
  3:   12,    54,     80,     39;
  4:   29,   174,    388,    381,     139;
  5:   70,   525,   1564,   2313,    1698,     495;
  6:  169,  1521,   5671,  11211,   12393,    7263,    1763;
  7:  408,  4284,  19184,  47490,   70184,   61920,   30196,    6279;
  8:  985, 11820,  61792, 183798,  340206,  401256,  294490,  122961,  22363;
  9: 2378, 32103, 191896, 666582, 1482828, 2190582, 2149072, 1350099, 492838, 79647;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (18*n^2-57*n+41+(120*n^2-384*n+273)*x+(334*n^2-1069*n+750)*x^2+(496*n^2-1575*n+1090)*x^3+(414*n^2-1296*n+885)*x^4+(184*n^2-565*n+381)*x^5+(34*n^2-102*n+68)*x^6)*P(n-1,x) - (24*n-52+(200*n-431)*x+(707*n-1511)*x^2+(1376*n-2910)*x^3+(1594*n-3330)*x^4+(1100*n-2267)*x^5+(419*n-851)*x^6+(68*n-136)*x^7)*d/dx P(n-1,x) + (8+80*x+345*x^2+839*x^3+1260*x^4+1198*x^5+705*x^6+235*x^7+34*x^8)*d^2/dx^2 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[((18*n^2 - 57*n + 41 + (120*n^2 - 384*n + 273)*x + (334*n^2 - 1069*n + 750)*x^2 + (496*n^2 - 1575*n + 1090)*x^3 + (414*n^2 - 1296*n + 885)*x^4 + (184*n^2 - 565*n + 381)*x^5 + (34*n^2 - 102*n + 68)*x^6)*P[n - 1, x] - (24*n - 52 + (200*n - 431)*x + (707*n - 1511)*x^2 + (1376*n - 2910)*x^3 + (1594*n - 3330)*x^4 + (1100*n - 2267)*x^5 + (419*n - 851)*x^6 + (68*n - 136)*x^7)*D[P[n - 1, x], x] + (8 + 80*x + 345*x^2 + 839*x^3 + 1260*x^4 + 1198*x^5 + 705*x^6 + 235*x^7 + 34*x^8)*D[P[n - 1, x], {x, 2}])/(n + 3*n*x + 3*n*x^2 + n*x^3)]];
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] = ((18*n^2 - 57*n + 41 + (120*n^2 - 384*n + 273)*x + (334*n^2 - 1069*n + 750)*x^2 + (496*n^2 - 1575*n + 1090)*x^3 + (414*n^2 - 1296*n + 885)*x^4 + (184*n^2 - 565*n + 381)*x^5 + (34*n^2 - 102*n + 68)*x^6)*v[i - 1] - (24*n - 52 + (200*n - 431)*x + (707*n - 1511)*x^2 + (1376*n - 2910)*x^3 + (1594*n - 3330)*x^4 + (1100*n - 2267)*x^5 + (419*n - 851)*x^6 + (68*n - 136)*x^7)*deriv(v[i - 1], x) + (8 + 80*x + 345*x^2 + 839*x^3 + 1260*x^4 + 1198*x^5 + 705*x^6 + 235*x^7 + 34*x^8)*deriv(deriv(v[i - 1], x), x))/(n + 3*n*x + 3*n*x^2 + n*x^3)); 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. [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