Real-rooted atlas

delannoy → basis A013609 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-4\,n x^{2} + 3\,n x^{3} + 2\,n x^{4} + 2\,n^{2} + 2\,n^{2} x + 6\,n^{2} x^{2} + 2\,n^{2} x^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-6 - 22\,x - 12\,x^{2} + 25\,x^{3} + 20\,x^{4} + 4\,x^{5} + 8\,n + 34\,n x + 38\,n x^{2} + 5\,n x^{3} - 2\,n x^{4} + 10\,n^{2} + 14\,n^{2} x - 10\,n^{2} x^{2} - 6\,n^{2} x^{3} - 4\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x + 38\,x^{2} + 87\,x^{3} + 85\,x^{4} + 32\,x^{5} + 4\,x^{6} + 4\,n + 28\,n x + 70\,n x^{2} + 74\,n x^{3} + 30\,n x^{4} + 4\,n x^{5} + 4\,n^{2} + 16\,n^{2} x + 18\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
21518437
348100568212
4145436424152161173
5420166223601464384326322
61183582210972996044969286433425
7326419240453925489636224128002176128174120

All 15 rows

Technical details
Node id
n-31b6df0e886e8c92 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-34de92a09eb7e034

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:     4,      2;
  2:    15,     18,      4;
  3:    48,    100,     56,       8;
  4:   145,    436,    424,     152,      16;
  5:   420,   1662,   2360,    1464,     384,     32;
  6:  1183,   5822,  10972,    9960,    4496,    928,     64;
  7:  3264,  19240,  45392,   54896,   36224,  12800,   2176,   128;
  8:  8865,  60936, 173072,  262992,  231456, 119296,  34560,  4992,   256;
  9: 23780, 186842, 621328, 1140304, 1263232, 868160, 365952, 89728, 11264, 512;
%F Formula
The row polynomials satisfy (2*n^2+(2*n^3+2*n^2)*x+(6*n^2-4*n)*x^2+(2*n^2+3*n)*x^3+2*n*x^4)*P(n,x) = -(4*n^3-10*n^2-8*n+6+(6*n^3-14*n^2-34*n+22)*x+(10*n^2-38*n+12)*x^2+(6*n^2-5*n-25)*x^3+(2*n-20)*x^4-4*x^5)*P(n-1,x) + (4*n^2+4*n+(16*n^2+28*n+6)*x+(18*n^2+70*n+38)*x^2+(4*n^2+74*n+87)*x^3+(30*n+85)*x^4+(4*n+32)*x^5+4*x^6)*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[(-(4*n^3 - 10*n^2 - 8*n + 6 + (6*n^3 - 14*n^2 - 34*n + 22)*x + (10*n^2 - 38*n + 12)*x^2 + (6*n^2 - 5*n - 25)*x^3 + (2*n - 20)*x^4 - 4*x^5)*P[n - 1, x] + (4*n^2 + 4*n + (16*n^2 + 28*n + 6)*x + (18*n^2 + 70*n + 38)*x^2 + (4*n^2 + 74*n + 87)*x^3 + (30*n + 85)*x^4 + (4*n + 32)*x^5 + 4*x^6)*D[P[n - 1, x], x])/(2*n^2 + (2*n^3 + 2*n^2)*x + (6*n^2 - 4*n)*x^2 + (2*n^2 + 3*n)*x^3 + 2*n*x^4)]];
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] = (-(4*n^3 - 10*n^2 - 8*n + 6 + (6*n^3 - 14*n^2 - 34*n + 22)*x + (10*n^2 - 38*n + 12)*x^2 + (6*n^2 - 5*n - 25)*x^3 + (2*n - 20)*x^4 - 4*x^5)*v[i - 1] + (4*n^2 + 4*n + (16*n^2 + 28*n + 6)*x + (18*n^2 + 70*n + 38)*x^2 + (4*n^2 + 74*n + 87)*x^3 + (30*n + 85)*x^4 + (4*n + 32)*x^5 + 4*x^6)*deriv(v[i - 1], x))/(2*n^2 + (2*n^3 + 2*n^2)*x + (6*n^2 - 4*n)*x^2 + (2*n^2 + 3*n)*x^3 + 2*n*x^4)); 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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.