Real-rooted atlas

A094638 → basis A013610 → D

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n x^{2} + 6\,n^{2} x + 18\,n^{2} x^{2} - n^{3} - 6\,n^{3} x - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-1 - 6\,n x + 2\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
133
2213657
31385224861146
4978592211826777626502
5766864872204201283176145800705717
666636730152318095168830567392060314928021402135
763892886399284844739614413107623981670021147415277157360730305540
867180321084278967470639722847496356648228258088099883286615692496211631616027733985820
97697808014470526881186196950855369266552160925827815298125838872343671557292225304530048642831033601161066124215
109556228802053626436819559903554810835407123923846531601665907271402893214214791490732142613746519688311142026080214277011200053149955546565
111278472896030950628249633637335127922160884611281690854433606150261195219773118519876988002666707237203474080629187238267272330455115633600777825550656002641883624459550
121834455340804944082157568604418749210804423742080806242153505711956670732044938883621417728600074505614305833082944155843682254083817477629463406033927680140971957949433603054733071667200141731682819120450
13281109128832083525901864192113541505511284893362164090248965171482261132380020327035876327915258128084823512633312183553480140684001857358223151930612200813112079589448014613408599847045126425378659154764801290624722779392008163609888207033225
14458278575552001489020196181760222933576441368162036264892216929281265873700643664040565548244105990680018679460302927573443461768109354970899848542766354094369108011679868389236184576011469577068054089265676594049209849517568311714099833232678405837594899956019200502530254282023035075
Technical details
Node id
n-c5c5d19ad588b4eb · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2708d889240ab808

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        3;
  2:       21,         36;
  3:      138,        522,         486;
  4:      978,       5922,       11826,        7776;
  5:     7668,      64872,      204201,      283176,       145800;
  6:    66636,     730152,     3180951,     6883056,      7392060,      3149280;
  7:   638928,    8639928,    48447396,   144131076,    239816700,    211474152,     77157360;
  8:  6718032,  108427896,   747063972,  2847496356,   6482282580,   8809988328,   6615692496,   2116316160;
  9: 76978080, 1447052688, 11861969508, 55369266552, 160925827815, 298125838872, 343671557292, 225304530048, 64283103360;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n^3+(6*n^3-6*n^2)*x+(9*n^3-18*n^2+9*n)*x^2)*P(n-2,x) + (2*n^2-1+(6*n^2-6*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 3.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 3;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*P[n - 2, x] + (2*n^2 - 1 + (6*n^2 - 6*n)*x)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 3; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + (6*n^3 - 6*n^2)*x + (9*n^3 - 18*n^2 + 9*n)*x^2)*v[i - 2] + (2*n^2 - 1 + (6*n^2 - 6*n)*x)*v[i - 1])/(n - 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.