Real-rooted atlas

A038320 → x ↦ x+1 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(289 + 510\,x + 144\,x^{2} + 306\,n x + 144\,n x^{2} + 36\,n^{2} x^{2}\right)\,P_{n}(x) = \left(4913 + 15606\,x + 10404\,x^{2} + 1944\,x^{3} + 5202\,n x + 5508\,n x^{2} + 1296\,n x^{3} + 612\,n^{2} x^{2} + 216\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 289 + 816\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02898163241429
149132080816524345645701
283521471648561816235008324001384393
3141985710022520159181209987840275400027993640382273
4241375692044594084059120603395865601404540002855347222861441145389213
54103386734055111592966070702810102700160557134200016989315842720511361791590431789098077
669757574417878502521621897602596827479344435218942562800077018231808184994772482436562944136048896867046201873
71185878764971506763607256478619028187270072328309765796424216800294594736665694347333964818639706521620815481088100776960023312839835609

All 15 rows

Technical details
Node id
n-2c65ee8ceabdf297 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-a775c45ff95fa257

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            289,             816,              324;
  1:           4913,           20808,            16524,             3456;
  2:          83521,          471648,           561816,           235008,            32400;
  3:        1419857,        10022520,         15918120,          9987840,          2754000,           279936;
  4:       24137569,       204459408,        405912060,        339586560,        140454000,         28553472,          2286144;
  5:      410338673,      4055111592,       9660707028,      10102700160,       5571342000,       1698931584,        272051136,        17915904;
  6:     6975757441,     78785025216,     218976025968,     274793444352,     189425628000,      77018231808,      18499477248,      2436562944,       136048896;
  7:   118587876497,   1506763607256,    4786190281872,    7007232830976,    5796424216800,    2945947366656,     943473339648,    186397065216,     20815481088,     1007769600;
  8:  2015993900449,  28461090359280,  101706543489780,  170175654466560,  164232019476000,  100162210466304,   40097616935040,  10562500362240,   1769315892480,   171320832000,    7316407296;
  9: 34271896307633, 532222389718536, 2113235959176540, 3977855923155840, 4387341091716000, 3121722226199808, 1499650873370496, 493796891934720, 110287357297920, 16018497792000, 1368168164352, 52242776064;
%F Formula
The row polynomials satisfy (289+(306*n+510)*x+(36*n^2+144*n+144)*x^2)*P(n,x) = (4913+(5202*n+15606)*x+(612*n^2+5508*n+10404)*x^2+(216*n^2+1296*n+1944)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 289 + 816*x + 324*x^2.
%t Mathematica
P[0, x_] := 289 + 816*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((4913 + (5202*n + 15606)*x + (612*n^2 + 5508*n + 10404)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*P[n - 1, x])/(289 + (306*n + 510)*x + (36*n^2 + 144*n + 144)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 289 + 816*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((4913 + (5202*n + 15606)*x + (612*n^2 + 5508*n + 10404)*x^2 + (216*n^2 + 1296*n + 1944)*x^3)*v[i - 1])/(289 + (306*n + 510)*x + (36*n^2 + 144*n + 144)*x^2)); 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

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.