Real-rooted atlas

A133087 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64 - 32\,x + 16\,x^{2} + 80\,n x - 72\,n x^{2} + 56\,n^{2} x^{2} - 16\,n^{2} x^{3} + n^{2} x^{4} + 12\,n^{3} x^{3} - 2\,n^{3} x^{4} + n^{4} x^{4}\right)\,P_{n}(x) = \left(512 - 48\,x - 96\,x^{2} - 108\,x^{3} - 16\,x^{4} + 2\,x^{5} - 320\,n + 1128\,n x + 80\,n x^{2} - 94\,n x^{3} - 32\,n x^{4} - n x^{5} - 680\,n^{2} x + 608\,n^{2} x^{2} + 258\,n^{2} x^{3} + 4\,n^{2} x^{4} - 4\,n^{2} x^{5} - 400\,n^{3} x^{2} + 12\,n^{3} x^{3} + 55\,n^{3} x^{4} + 2\,n^{3} x^{5} - 80\,n^{4} x^{3} - 22\,n^{4} x^{4} + 3\,n^{4} x^{5} - 5\,n^{5} x^{4} - 2\,n^{5} x^{5}\right)\,P_{n-1}(x) + \left(160 - 64\,x - 176\,x^{2} - 112\,x^{3} - 34\,x^{4} + x^{6} + 480\,n x + 104\,n x^{2} - 192\,n x^{3} - 86\,n x^{4} - 12\,n x^{5} + 440\,n^{2} x^{2} + 320\,n^{2} x^{3} + 28\,n^{2} x^{4} - 15\,n^{2} x^{5} - 2\,n^{2} x^{6} + 120\,n^{3} x^{3} + 104\,n^{3} x^{4} + 22\,n^{3} x^{5} + 10\,n^{4} x^{4} + 9\,n^{4} x^{5} + 2\,n^{4} x^{6}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
2416929
312646316155
41612821612825513
548448936768275362511
664768216025601500432497533
71922560820811520850034567356435235

All 15 rows

Technical details
Node id
n-e829ccd1ea4f8894 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d82a7263192d6c49

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,     4;
  2:   4,    16,     9;
  3:  12,    64,    63,     16;
  4:  16,   128,   216,    128,     25;
  5:  48,   448,   936,    768,    275,    36;
  6:  64,   768,  2160,   2560,   1500,   432,    49;
  7: 192,  2560,  8208,  11520,   8500,  3456,   735,    64;
  8: 256,  4096, 16128,  28672,  28000, 16128,  5488,  1024,   81;
  9: 768, 13312, 57600, 114688, 128800, 88704, 38416, 10240, 1539, 100;
%F Formula
The row polynomials satisfy (64+(80*n-32)*x+(56*n^2-72*n+16)*x^2+(12*n^3-16*n^2)*x^3+(n^4-2*n^3+n^2)*x^4)*P(n,x) = -(320*n-512+(680*n^2-1128*n+48)*x+(400*n^3-608*n^2-80*n+96)*x^2+(80*n^4-12*n^3-258*n^2+94*n+108)*x^3+(5*n^5+22*n^4-55*n^3-4*n^2+32*n+16)*x^4+(2*n^5-3*n^4-2*n^3+4*n^2+n-2)*x^5)*P(n-1,x) + (160+(480*n-64)*x+(440*n^2+104*n-176)*x^2+(120*n^3+320*n^2-192*n-112)*x^3+(10*n^4+104*n^3+28*n^2-86*n-34)*x^4+(9*n^4+22*n^3-15*n^2-12*n)*x^5+(2*n^4-2*n^2+1)*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[(-(320*n - 512 + (680*n^2 - 1128*n + 48)*x + (400*n^3 - 608*n^2 - 80*n + 96)*x^2 + (80*n^4 - 12*n^3 - 258*n^2 + 94*n + 108)*x^3 + (5*n^5 + 22*n^4 - 55*n^3 - 4*n^2 + 32*n + 16)*x^4 + (2*n^5 - 3*n^4 - 2*n^3 + 4*n^2 + n - 2)*x^5)*P[n - 1, x] + (160 + (480*n - 64)*x + (440*n^2 + 104*n - 176)*x^2 + (120*n^3 + 320*n^2 - 192*n - 112)*x^3 + (10*n^4 + 104*n^3 + 28*n^2 - 86*n - 34)*x^4 + (9*n^4 + 22*n^3 - 15*n^2 - 12*n)*x^5 + (2*n^4 - 2*n^2 + 1)*x^6)*D[P[n - 1, x], x])/(64 + (80*n - 32)*x + (56*n^2 - 72*n + 16)*x^2 + (12*n^3 - 16*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*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] = (-(320*n - 512 + (680*n^2 - 1128*n + 48)*x + (400*n^3 - 608*n^2 - 80*n + 96)*x^2 + (80*n^4 - 12*n^3 - 258*n^2 + 94*n + 108)*x^3 + (5*n^5 + 22*n^4 - 55*n^3 - 4*n^2 + 32*n + 16)*x^4 + (2*n^5 - 3*n^4 - 2*n^3 + 4*n^2 + n - 2)*x^5)*v[i - 1] + (160 + (480*n - 64)*x + (440*n^2 + 104*n - 176)*x^2 + (120*n^3 + 320*n^2 - 192*n - 112)*x^3 + (10*n^4 + 104*n^3 + 28*n^2 - 86*n - 34)*x^4 + (9*n^4 + 22*n^3 - 15*n^2 - 12*n)*x^5 + (2*n^4 - 2*n^2 + 1)*x^6)*deriv(v[i - 1], x))/(64 + (80*n - 32)*x + (56*n^2 - 72*n + 16)*x^2 + (12*n^3 - 16*n^2)*x^3 + (n^4 - 2*n^3 + n^2)*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.