Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A008277 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - x + x^{2} + 4\,n x - 4\,n x^{2} + 4\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x^{2} + 3\,x^{3} - 4\,n x^{3}\right)\,P_{n-1}'(x) + \left(x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
11315
21102515152
3121140350301631877
41364622646695177703025255121147
5155115511880639871794872467301457502850110231678570
6178243139325359502189961257154249321312750850125325302616254095127644437
711054550106470147947812662650671284902166278404087413334206932732107669204235595023751011638311382958545

All 15 rows

Technical details
Node id
n-cc5ac69d87f94be7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-6a8026fb20183be6

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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: 1,   3,     1;
  2: 1,  10,    25,     15,        1;
  3: 1,  21,   140,    350,      301,        63,          1;
  4: 1,  36,   462,   2646,     6951,      7770,       3025,         255,            1;
  5: 1,  55,  1155,  11880,    63987,    179487,     246730,      145750,        28501,         1023,             1;
  6: 1,  78,  2431,  39325,   359502,   1899612,    5715424,     9321312,      7508501,      2532530,        261625,          4095,             1;
  7: 1, 105,  4550, 106470,  1479478,  12662650,   67128490,   216627840,    408741333,    420693273,     210766920,      42355950,       2375101,        16383,            1;
  8: 1, 136,  7820, 249900,  4910178,  62022324,  512060978,  2758334150,   9528822303,  20415995028,   25708104786,   17505749898,    5652751651,    694337290,     21457825,       65535,         1;
  9: 1, 171, 12597, 527136, 13916778, 243577530, 2892439160, 23466951300, 129413217791, 477297033785, 1144614626805, 1709751003480, 1492924634839, 693081601779, 147589284710, 11259666950, 193448101, 262143, 1;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n-1)*x+(4*n^2-4*n+1)*x^2)*P(n-1,x) - (2*x^2+(4*n-3)*x^3)*d/dx P(n-1,x) + x^4*d^2/dx^2 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[(1 + (4*n - 1)*x + (4*n^2 - 4*n + 1)*x^2)*P[n - 1, x] - (2*x^2 + (4*n - 3)*x^3)*D[P[n - 1, x], x] + x^4*D[P[n - 1, x], {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] = (1 + (4*n - 1)*x + (4*n^2 - 4*n + 1)*x^2)*v[i - 1] - (2*x^2 + (4*n - 3)*x^3)*deriv(v[i - 1], x) + x^4*deriv(deriv(v[i - 1], x), x)); 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. [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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.