Real-rooted atlas

A122076 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-20 - 2\,x + 20\,n - 18\,n x + 4\,n x^{2} + 20\,n^{2} x - 12\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(-80 - 98\,x - 12\,x^{2} + 12\,x^{3} + 80\,n - 2\,n x - 92\,n x^{2} - 32\,n x^{3} + 100\,n^{2} x + 72\,n^{2} x^{2} - 4\,n^{2} x^{3} + 32\,n^{3} x^{2} + 24\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-25 - 50\,x - 19\,x^{2} + 14\,x^{3} + 8\,x^{4} - 50\,n x - 110\,n x^{2} - 76\,n x^{3} - 16\,n x^{4} - 20\,n^{2} x^{2} - 36\,n^{2} x^{3} - 16\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 3 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
022
1347
2716629
31860458131
4472082409610601
51236801065680175122735
632221364230372815752881412293
7843652415624175841039531924411654619

All 15 rows

Technical details
Node id
n-d57812740ab00556 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-14b3f6a4b38fc102

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:    2;
  1:    3,     4;
  2:    7,    16,      6;
  3:   18,    60,     45,      8;
  4:   47,   208,    240,     96,     10;
  5:  123,   680,   1065,    680,    175,     12;
  6:  322,  2136,   4230,   3728,   1575,    288,    14;
  7:  843,  6524,  15624,  17584,  10395,   3192,   441,    16;
  8: 2207, 19520,  54840,  75136,  57200,  24864,  5880,   640,  18;
  9: 5778, 57492, 185355, 299352, 279045, 157356, 53214, 10080, 891, 20;
%F Formula
The row polynomials satisfy (20*n-20+(20*n^2-18*n-2)*x+(8*n^3-12*n^2+4*n)*x^2)*P(n,x) = (80*n-80+(100*n^2-2*n-98)*x+(32*n^3+72*n^2-92*n-12)*x^2+(24*n^3-4*n^2-32*n+12)*x^3)*P(n-1,x) - (25+(50*n+50)*x+(20*n^2+110*n+19)*x^2+(36*n^2+76*n-14)*x^3+(16*n^2+16*n-8)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 2, P(1,x) = 3 + 4*x.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 3 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((80*n - 80 + (100*n^2 - 2*n - 98)*x + (32*n^3 + 72*n^2 - 92*n - 12)*x^2 + (24*n^3 - 4*n^2 - 32*n + 12)*x^3)*P[n - 1, x] - (25 + (50*n + 50)*x + (20*n^2 + 110*n + 19)*x^2 + (36*n^2 + 76*n - 14)*x^3 + (16*n^2 + 16*n - 8)*x^4)*D[P[n - 1, x], x])/(20*n - 20 + (20*n^2 - 18*n - 2)*x + (8*n^3 - 12*n^2 + 4*n)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 3 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((80*n - 80 + (100*n^2 - 2*n - 98)*x + (32*n^3 + 72*n^2 - 92*n - 12)*x^2 + (24*n^3 - 4*n^2 - 32*n + 12)*x^3)*v[i - 1] - (25 + (50*n + 50)*x + (20*n^2 + 110*n + 19)*x^2 + (36*n^2 + 76*n - 14)*x^3 + (16*n^2 + 16*n - 8)*x^4)*deriv(v[i - 1], x))/(20*n - 20 + (20*n^2 - 18*n - 2)*x + (8*n^3 - 12*n^2 + 4*n)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.