Real-rooted atlas

delannoy → basis A013611 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 6\,n x + 6\,n x^{2} + 2\,n x^{3}\right)\,P_{n}(x) = \left(22 + 180\,x + 570\,x^{2} + 920\,x^{3} + 810\,x^{4} + 372\,x^{5} + 70\,x^{6} - 27\,n - 222\,n x - 731\,n x^{2} - 1236\,n x^{3} - 1137\,n x^{4} - 542\,n x^{5} - 105\,n x^{6} + 9\,n^{2} + 66\,n^{2} x + 209\,n^{2} x^{2} + 356\,n^{2} x^{3} + 339\,n^{2} x^{4} + 170\,n^{2} x^{5} + 35\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(14 + 166\,x + 760\,x^{2} + 1800\,x^{3} + 2430\,x^{4} + 1894\,x^{5} + 796\,x^{6} + 140\,x^{7} - 6\,n - 70\,n x - 322\,n x^{2} - 778\,n x^{3} - 1082\,n x^{4} - 874\,n x^{5} - 382\,n x^{6} - 70\,n x^{7}\right)\,P_{n-1}'(x) + \left(1 + 16\,x + 102\,x^{2} + 340\,x^{3} + 660\,x^{4} + 776\,x^{5} + 546\,x^{6} + 212\,x^{7} + 35\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1268
25304176
312108308276704
4293481510279618616544
5701050612417316236941254660800
61693042222918497217747119240284581564928
74088568756083630001022984169092015173205702165249024

All 15 rows

Technical details
Node id
n-800db8dcc26dde3d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-07ec67562a37d493

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,     6;
  2:    5,    30,     41;
  3:   12,   108,    308,     276;
  4:   29,   348,   1510,    2796,     1861;
  5:   70,  1050,   6124,   17316,    23694,    12546;
  6:  169,  3042,  22291,   84972,   177471,   192402,     84581;
  7:  408,  8568,  75608,  363000,  1022984,  1690920,   1517320,    570216;
  8:  985, 23640, 244012, 1413576,  5022726, 11201832,  15306508,  11713848,  3844201;
  9: 2378, 64206, 758920, 5150712, 22105932, 62183844, 114601768, 133395480, 88979386, 25916286;
%F Formula
The row polynomials satisfy (2*n+6*n*x+6*n*x^2+2*n*x^3)*P(n,x) = (9*n^2-27*n+22+(66*n^2-222*n+180)*x+(209*n^2-731*n+570)*x^2+(356*n^2-1236*n+920)*x^3+(339*n^2-1137*n+810)*x^4+(170*n^2-542*n+372)*x^5+(35*n^2-105*n+70)*x^6)*P(n-1,x) - (6*n-14+(70*n-166)*x+(322*n-760)*x^2+(778*n-1800)*x^3+(1082*n-2430)*x^4+(874*n-1894)*x^5+(382*n-796)*x^6+(70*n-140)*x^7)*d/dx P(n-1,x) + (1+16*x+102*x^2+340*x^3+660*x^4+776*x^5+546*x^6+212*x^7+35*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[((9*n^2 - 27*n + 22 + (66*n^2 - 222*n + 180)*x + (209*n^2 - 731*n + 570)*x^2 + (356*n^2 - 1236*n + 920)*x^3 + (339*n^2 - 1137*n + 810)*x^4 + (170*n^2 - 542*n + 372)*x^5 + (35*n^2 - 105*n + 70)*x^6)*P[n - 1, x] - (6*n - 14 + (70*n - 166)*x + (322*n - 760)*x^2 + (778*n - 1800)*x^3 + (1082*n - 2430)*x^4 + (874*n - 1894)*x^5 + (382*n - 796)*x^6 + (70*n - 140)*x^7)*D[P[n - 1, x], x] + (1 + 16*x + 102*x^2 + 340*x^3 + 660*x^4 + 776*x^5 + 546*x^6 + 212*x^7 + 35*x^8)*D[P[n - 1, x], {x, 2}])/(2*n + 6*n*x + 6*n*x^2 + 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] = ((9*n^2 - 27*n + 22 + (66*n^2 - 222*n + 180)*x + (209*n^2 - 731*n + 570)*x^2 + (356*n^2 - 1236*n + 920)*x^3 + (339*n^2 - 1137*n + 810)*x^4 + (170*n^2 - 542*n + 372)*x^5 + (35*n^2 - 105*n + 70)*x^6)*v[i - 1] - (6*n - 14 + (70*n - 166)*x + (322*n - 760)*x^2 + (778*n - 1800)*x^3 + (1082*n - 2430)*x^4 + (874*n - 1894)*x^5 + (382*n - 796)*x^6 + (70*n - 140)*x^7)*deriv(v[i - 1], x) + (1 + 16*x + 102*x^2 + 340*x^3 + 660*x^4 + 776*x^5 + 546*x^6 + 212*x^7 + 35*x^8)*deriv(deriv(v[i - 1], x), x))/(2*n + 6*n*x + 6*n*x^2 + 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