Real-rooted atlas

A156992 → basis A013610 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(2 + 3\,n x\right)\,P_{n}(x) = \left(4 + 12\,x + 9\,x^{2} + 4\,n + 18\,n x + 18\,n x^{2} + 6\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
141216
224144162330
31921728388825928400
41920230407776010368048600255000
52304034560015552003110400291600010497609000000
6322560580608032659200870912001224720008817984025719120362250000
75160960108380160731566080243855360045722880004938071040288054144070543872016380000000

All 15 rows

Technical details
Node id
n-cf28bf26a0e97b2b · 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 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:          4,          12;
  2:         24,         144,          162;
  3:        192,        1728,         3888,          2592;
  4:       1920,       23040,        77760,        103680,         48600;
  5:      23040,      345600,      1555200,       3110400,       2916000,        1049760;
  6:     322560,     5806080,     32659200,      87091200,     122472000,       88179840,       25719120;
  7:    5160960,   108380160,    731566080,    2438553600,    4572288000,     4938071040,     2880541440,     705438720;
  8:   92897280,  2229534720,  17557585920,   70230343680,  164602368000,   237027409920,   207398983680,  101583175680,   21427701120;
  9: 1857945600, 50164531200, 451480780800, 2106910310400, 5925685248000, 10666233446400, 12443939020800, 9142485811200, 3856986201600, 714256704000;
%F Formula
The row polynomials satisfy (2+3*n*x)*P(n,x) = (4*n+4+(6*n^2+18*n+12)*x+(9*n^2+18*n+9)*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[((4*n + 4 + (6*n^2 + 18*n + 12)*x + (9*n^2 + 18*n + 9)*x^2)*P[n - 1, x])/(2 + 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] = ((4*n + 4 + (6*n^2 + 18*n + 12)*x + (9*n^2 + 18*n + 9)*x^2)*v[i - 1])/(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 - 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.