Real-rooted atlas

A395972 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(4 + 2\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-2 - 4\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 28 + 32\,x + 8\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
12832868
2272504288481112
3334487048064307238423568
449472167200217600134400384003840610912
585292835619846019200522240024192005529604608018674752
61675801683586944174537216196627200127948800474163209031680645120656551296
73690826242145026048534956441674469212166292070400327548928010115481601651507201032192026065174784
8899692236859791385088173747109888288876478464301600309248203863080960884382105602341011456033443020801857945601152253707776

All 15 rows

Technical details
Node id
n-6e47c52ce477d423 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-570ab0ccd1213152

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 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:            28,             32,               8;
   2:           272,            504,             288,              48;
   3:          3344,           8704,            8064,            3072,             384;
   4:         49472,         167200,          217600,          134400,           38400,            3840;
   5:        852928,        3561984,         6019200,         5222400,         2419200,          552960,           46080;
   6:      16758016,       83586944,       174537216,       196627200,       127948800,        47416320,         9031680,          645120;
   7:     369082624,     2145026048,      5349564416,      7446921216,      6292070400,      3275489280,      1011548160,       165150720,       10321920;
   8:    8996922368,    59791385088,    173747109888,    288876478464,    301600309248,    203863080960,     88438210560,     23410114560,     3344302080,      185794560;
   9:  240294124544,  1799384473600,   5979138508800,  11583140659200,  14443823923200,  12064012369920,   6795436032000,   2526806016000,   585252864000,    74317824000,    3715891200;
  10: 6974172532736, 58151178139648, 217725521305600, 482317173043200, 700780009881600, 699081077882880, 486581832253440, 234927931392000, 76435881984000, 15736799232000, 1798491340800, 81749606400;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+4+(4*n+2)*x)*P(n-1,x) - (2+4*x+2*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 28 + 32*x + 8*x^2.
%t Mathematica
P[1, x_] := 28 + 32*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[(4*n + 4 + (4*n + 2)*x)*P[n - 1, x] - (2 + 4*x + 2*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 28 + 32*x + 8*x^2; for(n = 2, N, my(i = n); v[i] = (4*n + 4 + (4*n + 2)*x)*v[i - 1] - (2 + 4*x + 2*x^2)*deriv(v[i - 1], 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.