Real-rooted atlas

A008278 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2\,x + n\right)\,P_{n}(x) = \left(-2\,x - 5\,x^{2} + 4\,x^{4} + 2\,n + 7\,n x + 11\,n x^{2} + 5\,n x^{3} - 2\,n x^{4} + n^{2} + n^{2} x - 3\,n^{2} x^{2} - 3\,n^{2} x^{3} - n^{3} - 2\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(x^{2} + 4\,x^{3} + 5\,x^{4} + 2\,x^{5} + n + 5\,n x + 10\,n x^{2} + 9\,n x^{3} + 3\,n x^{4} + n^{2} + 3\,n^{2} x + 3\,n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-4\,x - 5\,x^{2} + 2\,x^{3} - n + 2\,n x + 5\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-2\,x - 4\,x^{2} - 2\,x^{3} - n - 2\,n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 3 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2314
3107118
4384313195
5161261133231579
6747160612033684113966
737531015810253472897675129944
8202536652985475539731721625531411246141
911664245262371145757831525682660359667927112183173
107131303202057598181559976803468608115466920798517614529120744088
114607813235536195115415661331856441169151935272350260927127304705010431209903998
1231345921180028256446775250625711760541564063297999322103226194214924342445974127631206912250718875
132237672331428442744399508194664175944726519505417434395204619039220145353120709111859184415853519134119125467494151
1416714306071175283392536626944794665102124867779395866761136604391326567395381172118877227286732253853016402937204798578882171303014254098
15130301531181001547500753445325225536988883018429269256713828419577286175330901754692347301017567039458696913773433540163161497510319345228013321641113779215051492
Technical details
Node id
n-9bc7693e04afc93e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-39b0f76a8910e04d

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      1;
   2:      3,       1;
   3:     10,       7,       1;
   4:     38,      43,      13,       1;
   5:    161,     261,     133,      23,       1;
   6:    747,    1606,    1203,     368,      41,       1;
   7:   3753,   10158,   10253,    4728,     976,      75,      1;
   8:  20253,   66529,   85475,   53973,   17216,    2553,    141,     1;
   9: 116642,  452623,  711457,  578315,  256826,   60359,   6679,   271,   1;
  10: 713130, 3202057, 5981815, 5997680, 3468608, 1154669, 207985, 17614, 529, 1;
%F Formula
The row polynomials satisfy (n+2*x)*P(n,x) = -(n^3-n^2-2*n+(2*n^3-n^2-7*n+2)*x+(n^3+3*n^2-11*n+5)*x^2+(3*n^2-5*n)*x^3+(2*n-4)*x^4)*P(n-2,x) + (n^2+n+(3*n^2+5*n)*x+(3*n^2+10*n+1)*x^2+(n^2+9*n+4)*x^3+(3*n+5)*x^4+2*x^5)*d/dx P(n-2,x) + (2*n^2-n+(2*n^2+2*n-4)*x+(5*n-5)*x^2+2*x^3)*P(n-1,x) - (n+(2*n+2)*x+(n+4)*x^2+2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 3 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 3 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - n^2 - 2*n + (2*n^3 - n^2 - 7*n + 2)*x + (n^3 + 3*n^2 - 11*n + 5)*x^2 + (3*n^2 - 5*n)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] + (n^2 + n + (3*n^2 + 5*n)*x + (3*n^2 + 10*n + 1)*x^2 + (n^2 + 9*n + 4)*x^3 + (3*n + 5)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (2*n^2 - n + (2*n^2 + 2*n - 4)*x + (5*n - 5)*x^2 + 2*x^3)*P[n - 1, x] - (n + (2*n + 2)*x + (n + 4)*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 3 + x; for(n = 3, N, my(i = n); v[i] = (-(n^3 - n^2 - 2*n + (2*n^3 - n^2 - 7*n + 2)*x + (n^3 + 3*n^2 - 11*n + 5)*x^2 + (3*n^2 - 5*n)*x^3 + (2*n - 4)*x^4)*v[i - 2] + (n^2 + n + (3*n^2 + 5*n)*x + (3*n^2 + 10*n + 1)*x^2 + (n^2 + 9*n + 4)*x^3 + (3*n + 5)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (2*n^2 - n + (2*n^2 + 2*n - 4)*x + (5*n - 5)*x^2 + 2*x^3)*v[i - 1] - (n + (2*n + 2)*x + (n + 4)*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n + 2*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.