Real-rooted atlas

A038330 → basis A007318 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(4 + 2\,x + n x\right)\,P_{n}(x) = \left(64 + 64\,x + 12\,x^{2} + 16\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 256\,x + 48\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
025625648560
140966144230425612800
26553613107273728163841280288000
310485762621440196608065536010240061446400000
416777216503316484718592020971520491520058982428672140800000
526843545693952409610569646085872025601835008003303014432112641310723072000000
64294967296171798691842254857830415032385536587202560014092861442055208961677721658982466560000000
76871947673630923764531246385646796836077725286416911433728050734301184986500300812079595528493465626214401433600000000

All 15 rows

Technical details
Node id
n-defea092f45b33f9 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-af50e1602aec03e4

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:            256,            256,              48;
  1:           4096,           6144,            2304,             256;
  2:          65536,         131072,           73728,           16384,            1280;
  3:        1048576,        2621440,         1966080,          655360,          102400,           6144;
  4:       16777216,       50331648,        47185920,        20971520,         4915200,         589824,          28672;
  5:      268435456,      939524096,      1056964608,       587202560,       183500800,       33030144,        3211264,        131072;
  6:     4294967296,    17179869184,     22548578304,     15032385536,      5872025600,     1409286144,      205520896,      16777216,       589824;
  7:    68719476736,   309237645312,    463856467968,    360777252864,    169114337280,    50734301184,     9865003008,    1207959552,     84934656,     2621440;
  8:  1099511627776,  5497558138880,   9277129359360,   8246337208320,   4509715660800,  1623497637888,   394600120320,   64424509440,   6794772480,   419430400,   11534336;
  9: 17592186044416, 96757023244288, 181419418583040, 181419418583040, 113387136614400, 47622597378048, 13889924235264, 2834678415360, 398626652160, 36909875200, 2030043136, 50331648;
%F Formula
The row polynomials satisfy (4+(n+2)*x)*P(n,x) = (64+(16*n+64)*x+(4*n+12)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 256*x + 48*x^2.
%t Mathematica
P[0, x_] := 256 + 256*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((64 + (16*n + 64)*x + (4*n + 12)*x^2)*P[n - 1, x])/(4 + (n + 2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 256*x + 48*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((64 + (16*n + 64)*x + (4*n + 12)*x^2)*v[i - 1])/(4 + (n + 2)*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.