Real-rooted atlas

A143497 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
010212
190363129
28405048441432
3840067201680160516965
490720907203024043202706216276
510584001270080529200100800945042072968357
613305600186278409313920221760027720018480616843761264
7179625600287400960167650560479001607484400665280332648649690761097

All 15 rows

Technical details
Node id
n-0848a50e390b197e · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-1d70f6b910bc9524

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:          10,           2;
  1:          90,          36,           3;
  2:         840,         504,          84,           4;
  3:        8400,        6720,        1680,         160,          5;
  4:       90720,       90720,       30240,        4320,        270,         6;
  5:     1058400,     1270080,      529200,      100800,       9450,       420,        7;
  6:    13305600,    18627840,     9313920,     2217600,     277200,     18480,      616,       8;
  7:   179625600,   287400960,   167650560,    47900160,    7484400,    665280,    33264,     864,     9;
  8:  2594592000,  4670265600,  3113510400,  1037836800,  194594400,  21621600,  1441440,   56160,  1170,   10;
  9: 39956716800, 79913433600, 59935075200, 22832409600, 4994589600, 665945280, 55495440, 2882880, 90090, 1540, 11;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = (n^2+7*n+10+(n+2)*x)*P(n-1,x) + (n+2)*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 10 + 2*x.
%t Mathematica
P[0, x_] := 10 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + 7*n + 10 + (n + 2)*x)*P[n - 1, x] + (n + 2)*x*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 10 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((n^2 + 7*n + 10 + (n + 2)*x)*v[i - 1] + (n + 2)*x*deriv(v[i - 1], x))/(n + 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

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. [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.