Real-rooted atlas

A130534 → D²∘x (shift +1) → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-16 + 32\,x - 24\,x^{2} + 8\,x^{3} - x^{4} + 32\,n - 48\,n x + 24\,n x^{2} - 4\,n x^{3} - 24\,n^{2} + 24\,n^{2} x - 6\,n^{2} x^{2} + 8\,n^{3} - 4\,n^{3} x - n^{4}\right)\,P_{n-4}(x) + \left(-15 + 28\,x - 18\,x^{2} + 4\,x^{3} + 28\,n - 36\,n x + 12\,n x^{2} - 18\,n^{2} + 12\,n^{2} x + 4\,n^{3}\right)\,P_{n-3}(x) + \left(-7 + 12\,x - 6\,x^{2} + 12\,n - 12\,n x - 6\,n^{2}\right)\,P_{n-2}(x) + \left(-2 + 4\,x + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 6\,x\)
\(P_{2}(x) = 11 + 36\,x + 18\,x^{2}\)
\(P_{3}(x) = 50 + 210\,x + 180\,x^{2} + 40\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
211361865
35021018040480
427413501530600753829
517649744132307000157512633439
6130687879212184278400241503528196319976
710958470874412111128979603402006879670562883343740

All 15 rows

Technical details
Node id
n-37ab893ec74fdb40 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-26a71355ec76e3cb

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:        3,        6;
  2:       11,       36,        18;
  3:       50,      210,       180,        40;
  4:      274,     1350,      1530,       600,       75;
  5:     1764,     9744,     13230,      7000,     1575,      126;
  6:    13068,    78792,    121842,     78400,    24150,     3528,     196;
  7:   109584,   708744,   1211112,    897960,   340200,    68796,    7056,    288;
  8:  1026576,  7036200,  13026240,  10773000,  4745475,  1190700,  170520,  12960,   405;
  9: 10628640, 76521456, 151371000, 136677200, 67654125, 19879398, 3557400, 380160, 22275, 550;
%F Formula
The row polynomials satisfy P(n,x) = -(n^4-8*n^3+24*n^2-32*n+16+(4*n^3-24*n^2+48*n-32)*x+(6*n^2-24*n+24)*x^2+(4*n-8)*x^3+x^4)*P(n-4,x) + (4*n^3-18*n^2+28*n-15+(12*n^2-36*n+28)*x+(12*n-18)*x^2+4*x^3)*P(n-3,x) - (6*n^2-12*n+7+(12*n-12)*x+6*x^2)*P(n-2,x) + (4*n-2+4*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 3 + 6*x, P(2,x) = 11 + 36*x + 18*x^2, P(3,x) = 50 + 210*x + 180*x^2 + 40*x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 6*x;
P[2, x_] := 11 + 36*x + 18*x^2;
P[3, x_] := 50 + 210*x + 180*x^2 + 40*x^3;
P[n_, x_] := P[n, x] = Expand[-(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^3 - 24*n^2 + 48*n - 32)*x + (6*n^2 - 24*n + 24)*x^2 + (4*n - 8)*x^3 + x^4)*P[n - 4, x] + (4*n^3 - 18*n^2 + 28*n - 15 + (12*n^2 - 36*n + 28)*x + (12*n - 18)*x^2 + 4*x^3)*P[n - 3, x] - (6*n^2 - 12*n + 7 + (12*n - 12)*x + 6*x^2)*P[n - 2, x] + (4*n - 2 + 4*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 6*x; v[3] = 11 + 36*x + 18*x^2; v[4] = 50 + 210*x + 180*x^2 + 40*x^3; for(n = 4, N, my(i = n + 1); v[i] = -(n^4 - 8*n^3 + 24*n^2 - 32*n + 16 + (4*n^3 - 24*n^2 + 48*n - 32)*x + (6*n^2 - 24*n + 24)*x^2 + (4*n - 8)*x^3 + x^4)*v[i - 4] + (4*n^3 - 18*n^2 + 28*n - 15 + (12*n^2 - 36*n + 28)*x + (12*n - 18)*x^2 + 4*x^3)*v[i - 3] - (6*n^2 - 12*n + 7 + (12*n - 12)*x + 6*x^2)*v[i - 2] + (4*n - 2 + 4*x)*v[i - 1]); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.