Real-rooted atlas

A136572 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1167
22245480
361084866481248
424576388810368972024576
5120360032400129600243000174960583680
672025920291600155520043740006298560367416016220160
750402116802857680190512007144200015431472018003384088179840516096000

All 15 rows

Technical details
Node id
n-c4e421ea1b8043a3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-3151c955f16eaf46

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,        6;
  2:      2,       24,        54;
  3:      6,      108,       486,        648;
  4:     24,      576,      3888,      10368,        9720;
  5:    120,     3600,     32400,     129600,      243000,      174960;
  6:    720,    25920,    291600,    1555200,     4374000,     6298560,      3674160;
  7:   5040,   211680,   2857680,   19051200,    71442000,   154314720,    180033840,     88179840;
  8:  40320,  1935360,  30481920,  243855360,  1143072000,  3292047360,   5761082880,   5643509760,   2380855680;
  9: 362880, 19595520, 352719360, 3292047360, 18517766400, 66663959040, 155549237760, 228562145280, 192849310080, 71425670400;
%F Formula
The row polynomials satisfy (1+3*n*x)*P(n,x) = (n+(3*n^2+6*n)*x+(9*n^2+9*n)*x^2)*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[Together[((n + (3*n^2 + 6*n)*x + (9*n^2 + 9*n)*x^2)*P[n - 1, x])/(1 + 3*n*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 + (3*n^2 + 6*n)*x + (9*n^2 + 9*n)*x^2)*v[i - 1])/(1 + 3*n*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

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.