Real-rooted atlas

A145596 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(64\,n - 32\,n x + 80\,n^{2} + 48\,n^{2} x + 32\,n^{3} + 98\,n^{3} x + 16\,n^{3} x^{2} + 4\,n^{4} + 45\,n^{4} x + 20\,n^{4} x^{2} + 6\,n^{5} x + 8\,n^{5} x^{2} + n^{6} x^{2}\right)\,P_{n}(x) = \left(96 + 80\,x + 40\,x^{2} + 8\,x^{3} + 256\,n + 452\,n x + 194\,n x^{2} + 42\,n x^{3} + 248\,n^{2} + 778\,n^{2} x + 452\,n^{2} x^{2} + 92\,n^{2} x^{3} + 104\,n^{3} + 574\,n^{3} x + 500\,n^{3} x^{2} + 113\,n^{3} x^{3} + 16\,n^{4} + 192\,n^{4} x + 254\,n^{4} x^{2} + 77\,n^{4} x^{3} + 24\,n^{5} x + 56\,n^{5} x^{2} + 25\,n^{5} x^{3} + 4\,n^{6} x^{2} + 3\,n^{6} x^{3}\right)\,P_{n-1}(x) + \left(-32\,x - 60\,x^{2} - 36\,x^{3} - 8\,x^{4} - 64\,n x - 179\,n x^{2} - 149\,n x^{3} - 34\,n x^{4} - 40\,n^{2} x - 187\,n^{2} x^{2} - 205\,n^{2} x^{3} - 58\,n^{2} x^{4} - 8\,n^{3} x - 80\,n^{3} x^{2} - 118\,n^{3} x^{3} - 46\,n^{3} x^{4} - 12\,n^{4} x^{2} - 28\,n^{4} x^{3} - 16\,n^{4} x^{4} - 2\,n^{5} x^{3} - 2\,n^{5} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14812
214562797
34828828864688
4165132020259601254595
55725720118808800250021629688
62002240246306364064288755544343187915
770729900831449640768025480078624109765121173168

All 15 rows

Technical details
Node id
n-4961a6d753520c9c · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-b02aad238b7968d2

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,       8;
  2:    14,      56,      27;
  3:    48,     288,     288,       64;
  4:   165,    1320,    2025,      960,      125;
  5:   572,    5720,   11880,     8800,     2500,     216;
  6:  2002,   24024,   63063,    64064,    28875,    5544,     343;
  7:  7072,   99008,  314496,   407680,   254800,   78624,   10976,    512;
  8: 25194,  403104, 1503684,  2376192,  1911000,  825552,  187278,  19968,   729;
  9: 90440, 1627920, 6976800, 13023360, 12852000, 7197120, 2304960, 403200, 34020, 1000;
%F Formula
The row polynomials satisfy (4*n^4+32*n^3+80*n^2+64*n+(6*n^5+45*n^4+98*n^3+48*n^2-32*n)*x+(n^6+8*n^5+20*n^4+16*n^3)*x^2)*P(n,x) = (16*n^4+104*n^3+248*n^2+256*n+96+(24*n^5+192*n^4+574*n^3+778*n^2+452*n+80)*x+(4*n^6+56*n^5+254*n^4+500*n^3+452*n^2+194*n+40)*x^2+(3*n^6+25*n^5+77*n^4+113*n^3+92*n^2+42*n+8)*x^3)*P(n-1,x) - ((8*n^3+40*n^2+64*n+32)*x+(12*n^4+80*n^3+187*n^2+179*n+60)*x^2+(2*n^5+28*n^4+118*n^3+205*n^2+149*n+36)*x^3+(2*n^5+16*n^4+46*n^3+58*n^2+34*n+8)*x^4)*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[((16*n^4 + 104*n^3 + 248*n^2 + 256*n + 96 + (24*n^5 + 192*n^4 + 574*n^3 + 778*n^2 + 452*n + 80)*x + (4*n^6 + 56*n^5 + 254*n^4 + 500*n^3 + 452*n^2 + 194*n + 40)*x^2 + (3*n^6 + 25*n^5 + 77*n^4 + 113*n^3 + 92*n^2 + 42*n + 8)*x^3)*P[n - 1, x] - ((8*n^3 + 40*n^2 + 64*n + 32)*x + (12*n^4 + 80*n^3 + 187*n^2 + 179*n + 60)*x^2 + (2*n^5 + 28*n^4 + 118*n^3 + 205*n^2 + 149*n + 36)*x^3 + (2*n^5 + 16*n^4 + 46*n^3 + 58*n^2 + 34*n + 8)*x^4)*D[P[n - 1, x], x])/(4*n^4 + 32*n^3 + 80*n^2 + 64*n + (6*n^5 + 45*n^4 + 98*n^3 + 48*n^2 - 32*n)*x + (n^6 + 8*n^5 + 20*n^4 + 16*n^3)*x^2)]];
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] = ((16*n^4 + 104*n^3 + 248*n^2 + 256*n + 96 + (24*n^5 + 192*n^4 + 574*n^3 + 778*n^2 + 452*n + 80)*x + (4*n^6 + 56*n^5 + 254*n^4 + 500*n^3 + 452*n^2 + 194*n + 40)*x^2 + (3*n^6 + 25*n^5 + 77*n^4 + 113*n^3 + 92*n^2 + 42*n + 8)*x^3)*v[i - 1] - ((8*n^3 + 40*n^2 + 64*n + 32)*x + (12*n^4 + 80*n^3 + 187*n^2 + 179*n + 60)*x^2 + (2*n^5 + 28*n^4 + 118*n^3 + 205*n^2 + 149*n + 36)*x^3 + (2*n^5 + 16*n^4 + 46*n^3 + 58*n^2 + 34*n + 8)*x^4)*deriv(v[i - 1], x))/(4*n^4 + 32*n^3 + 80*n^2 + 64*n + (6*n^5 + 45*n^4 + 98*n^3 + 48*n^2 - 32*n)*x + (n^6 + 8*n^5 + 20*n^4 + 16*n^3)*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.