Real-rooted atlas

A143497 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x + n x\right)\,P_{n}(x) = \left(10 + 14\,x + 3\,x^{2} + 2\,n + 7\,n x + n x^{2} + n^{2} x\right)\,P_{n-1}(x) + \left(2\,x + 3\,x^{2} + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 20 + 20\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02020343
1120180544358
2840168075611253393
3672016800100802240200636046
4604801814401360804032054003247424051
5604800211680019051207056001260001134049085470158
66652800266112002794176012418560277200033264021560704976751233
77983360035925120043110144022353408059875200898128077616038016972101163391958

All 15 rows

Technical details
Node id
n-106a2e627392653c · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-2886ec3559ef5b58

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:          20,          20,            3;
  1:         120,         180,           54,           4;
  2:         840,        1680,          756,         112,           5;
  3:        6720,       16800,        10080,        2240,         200,          6;
  4:       60480,      181440,       136080,       40320,        5400,        324,         7;
  5:      604800,     2116800,      1905120,      705600,      126000,      11340,       490,        8;
  6:     6652800,    26611200,     27941760,    12418560,     2772000,     332640,     21560,      704,       9;
  7:    79833600,   359251200,    431101440,   223534080,    59875200,    8981280,    776160,    38016,     972,     10;
  8:  1037836800,  5189184000,   7005398400,  4151347200,  1297296000,  233513280,  25225200,  1647360,   63180,   1300,   11;
  9: 14529715200, 79913433600, 119870150400, 79913433600, 28540512000, 5993507520, 776936160, 63423360, 3243240, 100100, 1694, 12;
%F Formula
The row polynomials satisfy (2+(n+2)*x)*P(n,x) = (2*n+10+(n^2+7*n+14)*x+(n+3)*x^2)*P(n-1,x) + (2*x+(n+3)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 20 + 20*x + 3*x^2.
%t Mathematica
P[0, x_] := 20 + 20*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 10 + (n^2 + 7*n + 14)*x + (n + 3)*x^2)*P[n - 1, x] + (2*x + (n + 3)*x^2)*D[P[n - 1, x], x])/(2 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 20 + 20*x + 3*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2*n + 10 + (n^2 + 7*n + 14)*x + (n + 3)*x^2)*v[i - 1] + (2*x + (n + 3)*x^2)*deriv(v[i - 1], x))/(2 + (n + 2)*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

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