Real-rooted atlas

A075497 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-7 + 2\,x + x^{2}\right)\,P_{n-2}(x) + \left(-2 + 2\,x^{2}\right)\,P_{n-2}'(x) + \left(6\right)\,P_{n-1}(x) + \left(2 + 2\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
21910130
310485181208
46417162382811624
543806221286053040114032
632803565263345186201025541132480
72667685397853922941309912170017997011353408
8233750554235764681916193418441648648104294088114844800
921925236573021375731816828345108758217611395029693645481081173713920
10218946003635672082722905621417442600134304430251223002780442181320673513012157361664
11231693925673904619819413547770622588462123510938005303492267310041261972023194409625154128307903488
122587859331389874451876126687424810944959380204108989787110914058056181633870019172656812833007535700133541801390961811456
13304020964876114099176781317624785212841464152245062721858626632221568166391434950893525545123012461815068250005478620041807020815664598200320
143745210267939150937345518942534128748223923204504205428128075875000714474232180418101720338219515308688709615394431481103595692246259213133915623933238185853324468224
Technical details
Node id
n-6d48db30e2c09f75 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-96d1aa294925441a

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:        4,        1;
  2:       19,       10,        1;
  3:      104,       85,       18,        1;
  4:      641,      716,      238,       28,       1;
  5:     4380,     6221,     2860,      530,      40,       1;
  6:    32803,    56526,    33451,     8620,    1025,      54,     1;
  7:   266768,   539785,   392294,   130991,   21700,    1799,    70,    1;
  8:  2337505,  5423576,  4681916,  1934184,  416486,   48104,  2940,   88,   1;
  9: 21925236, 57302137, 57318168, 28345108, 7582176, 1139502, 96936, 4548, 108, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(7-2*x-x^2)*P(n-2,x) - (2-2*x^2)*d/dx P(n-2,x) + 6*P(n-1,x) + (2+2*x)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[-(7 - 2*x - x^2)*P[n - 2, x] - (2 - 2*x^2)*D[P[n - 2, x], x] + 6*P[n - 1, x] + (2 + 2*x)*D[P[n - 1, x], 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] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -(7 - 2*x - x^2)*v[i - 2] - (2 - 2*x^2)*deriv(v[i - 2], x) + 6*v[i - 1] + (2 + 2*x)*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

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.