Real-rooted atlas

A136124 → θ+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
2610319
32452274107
4120308213565702
57202088174062010065274
65040160561531266601475162744712
740320138528146580736962012530662458422568
83628801327392152701285870427324551744578235294407120
936288001399968017261208106230563805830845334118188101284861050292720
1039916800161254080210872808139868560551429501386567022862912464801674065011623471040
1147900160020148825602772354816195958646483655110023255958043612107557066447817026400847128344624320
1262270208002715147648039062936448291710971201332464738040271358608382769011222610401248320787450040051108013119938250880
138717829120039257471232058760832499246047927494422300893472072369478896164342017842669685304312308568261132301522664588121352141842662908800
141307674368000606297726720094029869433607690666890816392073311448013531529051043309440854725882722445676880244877387079705156451325432688399316661530136443724800
Technical details
Node id
n-c86aa45832718393 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-13dd874d79e2c744

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:       1;
  1:       2,        2;
  2:       6,       10,        3;
  3:      24,       52,       27,        4;
  4:     120,      308,      213,       56,       5;
  5:     720,     2088,     1740,      620,     100,      6;
  6:    5040,    16056,    15312,     6660,    1475,    162,      7;
  7:   40320,   138528,   146580,    73696,   20125,   3066,    245,     8;
  8:  362880,  1327392,  1527012,   858704,  273245,  51744,   5782,   352,   9;
  9: 3628800, 13999680, 17261208, 10623056, 3805830, 845334, 118188, 10128, 486, 10;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n*x+x^2)*P(n-2,x) + (2*n+1+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n*x + x^2)*P[n - 2, x] + (2*n + 1 + 2*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; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n*x + x^2)*v[i - 2] + (2*n + 1 + 2*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.