Real-rooted atlas

delannoy → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(n + 3\,n x + 3\,n x^{2} + n x^{3}\right)\,P_{n}(x) = \left(19 + 137\,x + 400\,x^{2} + 610\,x^{3} + 515\,x^{4} + 229\,x^{5} + 42\,x^{6} - 25\,n - 183\,n x - 547\,n x^{2} - 856\,n x^{3} - 741\,n x^{4} - 337\,n x^{5} - 63\,n x^{6} + 8\,n^{2} + 56\,n^{2} x + 165\,n^{2} x^{2} + 260\,n^{2} x^{3} + 230\,n^{2} x^{4} + 108\,n^{2} x^{5} + 21\,n^{2} x^{6}\right)\,P_{n-1}(x) + \left(18 + 171\,x + 669\,x^{2} + 1410\,x^{3} + 1740\,x^{4} + 1263\,x^{5} + 501\,x^{6} + 84\,x^{7} - 8\,n - 76\,n x - 300\,n x^{2} - 642\,n x^{3} - 808\,n x^{4} - 600\,n x^{5} - 244\,n x^{6} - 42\,n x^{7}\right)\,P_{n-1}'(x) + \left(2 + 24\,x + 121\,x^{2} + 336\,x^{3} + 565\,x^{4} + 592\,x^{5} + 379\,x^{6} + 136\,x^{7} + 21\,x^{8}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
25201944
3127214088312
4292326828724092224
570700275653365078190015840
61692028100092599237443283648827112832
740857123389611048021354424472015394441008803712

All 15 rows

Technical details
Node id
n-266498645a09d3d0 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-c569cb6e352750ae

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:    5,    20,     19;
  3:   12,    72,    140,      88;
  4:   29,   232,    682,     872,     409;
  5:   70,   700,   2756,    5336,    5078,    1900;
  6:  169,  2028,  10009,   25992,   37443,   28364,     8827;
  7:  408,  5712,  33896,  110480,  213544,  244720,   153944,   41008;
  8:  985, 15760, 109268,  428656, 1040446, 1599728,  1521380,  818192,  190513;
  9: 2378, 42804, 339544, 1557584, 4552668, 8791576, 11215128, 9112848, 4279658, 885076;
%F Formula
The row polynomials satisfy (n+3*n*x+3*n*x^2+n*x^3)*P(n,x) = (8*n^2-25*n+19+(56*n^2-183*n+137)*x+(165*n^2-547*n+400)*x^2+(260*n^2-856*n+610)*x^3+(230*n^2-741*n+515)*x^4+(108*n^2-337*n+229)*x^5+(21*n^2-63*n+42)*x^6)*P(n-1,x) - (8*n-18+(76*n-171)*x+(300*n-669)*x^2+(642*n-1410)*x^3+(808*n-1740)*x^4+(600*n-1263)*x^5+(244*n-501)*x^6+(42*n-84)*x^7)*d/dx P(n-1,x) + (2+24*x+121*x^2+336*x^3+565*x^4+592*x^5+379*x^6+136*x^7+21*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[((8*n^2 - 25*n + 19 + (56*n^2 - 183*n + 137)*x + (165*n^2 - 547*n + 400)*x^2 + (260*n^2 - 856*n + 610)*x^3 + (230*n^2 - 741*n + 515)*x^4 + (108*n^2 - 337*n + 229)*x^5 + (21*n^2 - 63*n + 42)*x^6)*P[n - 1, x] - (8*n - 18 + (76*n - 171)*x + (300*n - 669)*x^2 + (642*n - 1410)*x^3 + (808*n - 1740)*x^4 + (600*n - 1263)*x^5 + (244*n - 501)*x^6 + (42*n - 84)*x^7)*D[P[n - 1, x], x] + (2 + 24*x + 121*x^2 + 336*x^3 + 565*x^4 + 592*x^5 + 379*x^6 + 136*x^7 + 21*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] = ((8*n^2 - 25*n + 19 + (56*n^2 - 183*n + 137)*x + (165*n^2 - 547*n + 400)*x^2 + (260*n^2 - 856*n + 610)*x^3 + (230*n^2 - 741*n + 515)*x^4 + (108*n^2 - 337*n + 229)*x^5 + (21*n^2 - 63*n + 42)*x^6)*v[i - 1] - (8*n - 18 + (76*n - 171)*x + (300*n - 669)*x^2 + (642*n - 1410)*x^3 + (808*n - 1740)*x^4 + (600*n - 1263)*x^5 + (244*n - 501)*x^6 + (42*n - 84)*x^7)*deriv(v[i - 1], x) + (2 + 24*x + 121*x^2 + 336*x^3 + 565*x^4 + 592*x^5 + 379*x^6 + 136*x^7 + 21*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