Real-rooted atlas

A173018 → x ↦ x+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
111
211
3314
4138122
57567171160
65416342463411456
746836793356279867115904
84729381780541111690424511321202672
9545835109596787625535237173649734126112951680
10708726116205198151963487487306205669030614621748518148361216
11102247563262283125282270272164793976562592061124705812390806419210311880405504
12162263257346144907925605838349378561624015474201063901686245905384249398241894652056117630351872
13280915675958771542074711873046241391071811653434048648461324464363025671234183021174181953282356015141051385148108800
145268583483811791916217922267431754414622972146918101251463111575448879874052106452725484162986850121520037195769288901659906820219114999832576
15106413429704433915787758249763880374891270607556525455863725529485120515372293786991432093193160481864059334010096750163775669878553025553984930482163951232311251144704
Technical details
Node id
n-52c11a3a9ef4e2e5 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-6c43fbf0495b302d

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:       1;
   3:       3,        1;
   4:      13,        8,        1;
   5:      75,       67,       17,       1;
   6:     541,      634,      246,      34,       1;
   7:    4683,     6793,     3562,     798,      67,      1;
   8:   47293,    81780,    54111,   16904,    2451,    132,     1;
   9:  545835,  1095967,   876255,  352371,   73649,   7341,   261,   1;
  10: 7087261, 16205198, 15196348, 7487306, 2056690, 306146, 21748, 518, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-3+(n-2)*x)*P(n-1,x) - (2+3*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(2*n - 3 + (n - 2)*x)*P[n - 1, x] - (2 + 3*x + x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (2*n - 3 + (n - 2)*x)*v[i - 1] - (2 + 3*x + x^2)*deriv(v[i - 1], 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.