Real-rooted atlas

A085880 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2\,n - 3\,n x + 3\,n^{2} x\right)\,P_{n}(x) = \left(-24 - 36\,x + 16\,n - 12\,n x - 54\,n x^{2} + 24\,n^{2} x + 36\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2268
384854110
4403608105401750
52242688907212096567029750
61344201609072018144017010061236525000
784481520648553602280960320760023094726735969487500
85491211531527783776259459204864860052540488306486187505784174281250
9366080878592069189120276756480648648000934053120817296480400308480844400703239843750
1024893446721228860491059228229160967939451520142910127361667284819212249439488516773228495698746060775000000
11171991045159731205224227840278625484809142398720019747581235228798555968028210830336017852166072066119133600109096570441148164062500
1212039372839729930244469617152026817702912010056638592002534272925184443497761907254305848396804582055958480254558664360084004359238812496516250421815117187500
1385201715230672617472379573641216253049094144010675508659200307454649384966277199091609692236394815488972805726569607205968344960035669543307552106124261080321437099368796416430273437500
1460858368002373476352003204193075200234974158848001101441369600003568670037504008326896754176001427468015001600180663920648640016728140800800001104057292852800492719783587200133444941388200165819157938007980371093750000
15438180249601840357048320269152218316802153217746534401110252900556800399691044200448010491889910261760205555394160230403035153866897152033723931854412800278222437798905601655538472852992067256250459652801671457112015040191876454185400153440771484375000
Technical details
Node id
n-1866605ebe851ad2 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-311883d116e971c2

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 -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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:       1;
   2:       2,        6;
   3:       8,       48,        54;
   4:      40,      360,       810,        540;
   5:     224,     2688,      9072,      12096,       5670;
   6:    1344,    20160,     90720,     181440,     170100,       61236;
   7:    8448,   152064,    855360,    2280960,    3207600,     2309472,      673596;
   8:   54912,  1153152,   7783776,   25945920,   48648600,    52540488,    30648618,     7505784;
   9:  366080,  8785920,  69189120,  276756480,  648648000,   934053120,   817296480,   400308480,   84440070;
  10: 2489344, 67212288, 604910592, 2822916096, 7939451520, 14291012736, 16672848192, 12249439488, 5167732284, 956987460;
%F Formula
The row polynomials satisfy (2*n+(3*n^2-3*n)*x)*P(n,x) = (16*n-24+(24*n^2-12*n-36)*x+(36*n^2-54*n)*x^2)*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[Together[((16*n - 24 + (24*n^2 - 12*n - 36)*x + (36*n^2 - 54*n)*x^2)*P[n - 1, x])/(2*n + (3*n^2 - 3*n)*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] = ((16*n - 24 + (24*n^2 - 12*n - 36)*x + (36*n^2 - 54*n)*x^2)*v[i - 1])/(2*n + (3*n^2 - 3*n)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.