Real-rooted atlas

A008955 → basis A007318 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
24131027
336157220100513
4576312461895340170016929
5144009307623594929441418134044200863379
651840038835361193797619329017174215588300780163540063026667
725401600216213264779137624154402063318201071921277816120495173600817700006239640033

All 15 rows

Technical details
Node id
n-6918d76cf43c5e24 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ce583c2d05188b3b

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 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            1;
  1:            1,             2;
  2:            4,            13,            10;
  3:           36,           157,           220,            100;
  4:          576,          3124,          6189,           5340,           1700;
  5:        14400,         93076,        235949,         294414,         181340,          44200;
  6:       518400,       3883536,      11937976,       19329017,       17421558,        8300780,        1635400;
  7:     25401600,     216213264,     779137624,     1544020633,     1820107192,     1277816120,      495173600,       81770000;
  8:   1625702400,   15488752896,   63918670096,   149461266072,   216848201433,   200087199160,   114749158200,    37419564000,    5315050000;
  9: 131681894400, 1387896581376, 6447490015248, 17347693499704, 29820528133977, 33988615649466, 25701832145320, 12440415656400, 3498923298000, 435834100000;
%F Formula
The row polynomials satisfy P(n,x) = (n^2+(n^2+1)*x)*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[(n^2 + (n^2 + 1)*x)*P[n - 1, x]];
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] = (n^2 + (n^2 + 1)*x)*v[i - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.