Real-rooted atlas

A075502 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + x\right)\,P_{n-1}(x) + \left(7 + 7\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 71 + 23\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
07123195
17294304511205
288427809144174118167
31253991461514106536151101316341
4202624928925741125496148980761015316201063
53645801061171803307820665658996428295142312031134513605
671986670113849839798572161712068852102157570110487542443026013191601207
71545382146133495652780245883291517461006889102031683667698183228299439306324182089147925

All 15 rows

Technical details
Node id
n-2b086b257119fd8b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-82f087c518e57ff3

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:            71,             23,              1;
  1:           729,            430,             45,             1;
  2:          8842,           7809,           1441,            74,             1;
  3:        125399,         146151,          41065,          3615,           110,            1;
  4:       2026249,        2892574,        1125496,        148980,          7610,          153,           1;
  5:      36458010,       61171803,       30782066,       5658996,        428295,        14231,         203,         1;
  6:     719866701,     1384983979,      857216171,     206885210,      21575701,      1048754,       24430,       260,       1;
  7:   15453821461,    33495652780,    24588329151,    7461006889,    1020316836,     67698183,     2282994,     39306,     324,     1;
  8:  358100141148,   862125221275,   731120038771,  269526400340,   46561849390,   4027224453,   183773877,   4541580,   60105,   395,   1;
  9: 8899677678109, 23525659003067, 22606820481377, 9851117431551, 2086705834235, 227451003703, 13438455723, 446009817, 8413185, 88220, 473, 1;
%F Formula
The row polynomials satisfy P(n,x) = (8+x)*P(n-1,x) + (7+7*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 71 + 23*x + x^2.
%t Mathematica
P[0, x_] := 71 + 23*x + x^2;
P[n_, x_] := P[n, x] = Expand[(8 + x)*P[n - 1, x] + (7 + 7*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 71 + 23*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (8 + x)*v[i - 1] + (7 + 7*x)*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 - 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.