Real-rooted atlas

A241171 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + 6\,x\right)\,P_{n-1}(x) + \left(12 + 21\,x + 9\,x^{2}\right)\,P_{n-1}'(x) + \left(3 + 8\,x + 7\,x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17613
212121090421
33907102068820252025453
42027417073709204305292001134002473141
51543020767365606117132840101372040436590007484400352444093
616191957618488828530184768283702136934800013848634800476756280068108040069251478661
7224061282907137112377220635849487000605190975349560449529280200023279328072006674587920008172964800017943523153933

All 15 rows

Technical details
Node id
n-facdd49d404b663a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-073a6777529fcec8

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:                1;
  1:                7,                 6;
  2:              121,               210,                 90;
  3:             3907,             10206,               8820,               2520;
  4:           202741,            707370,             920430,             529200,             113400;
  5:         15430207,          67365606,          117132840,          101372040,           43659000,            7484400;
  6:       1619195761,        8488828530,        18476828370,        21369348000,        13848634800,         4767562800,          681080400;
  7:     224061282907,     1371123772206,      3584948700060,      5190975349560,      4495292802000,      2327932807200,       667458792000,        81729648000;
  8:   39531606447181,   276571337613690,    844291094921910,   1468765934672400,   1592486639944200,   1101873471098400,    475106703271200,    116709937344000,    12504636144000;
  9: 8661323866026007, 68190617712846006, 238044766594758480, 483572419758991080, 629951395986549000, 545714976479072400, 314351037648648000, 116101378384992000, 24946749107280000, 2375880867360000;
%F Formula
The row polynomials satisfy P(n,x) = (7+6*x)*P(n-1,x) + (12+21*x+9*x^2)*d/dx P(n-1,x) + (3+8*x+7*x^2+2*x^3)*d^2/dx^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[(7 + 6*x)*P[n - 1, x] + (12 + 21*x + 9*x^2)*D[P[n - 1, x], x] + (3 + 8*x + 7*x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}]];
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] = (7 + 6*x)*v[i - 1] + (12 + 21*x + 9*x^2)*deriv(v[i - 1], x) + (3 + 8*x + 7*x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.