Real-rooted atlas

A075499 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(5 + x\right)\,P_{n-1}(x) + \left(4 + 4\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 29 + 14\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
02914144
1201163271392
2165718845264414112
315821228219250129065149248
4170389295210158551322202675901659136
520327854095687274301374909190195494911919715840
626546673608380724874394816920680274216621677684281521156016896
73760856539640409058975575563802701647884168683638384654441347618912705638912

All 15 rows

Technical details
Node id
n-8680abea4735117b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4f6fdfe6deb1db49

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:          29,           14,            1;
  1:         201,          163,           27,            1;
  2:        1657,         1884,          526,           44,           1;
  3:       15821,        22821,         9250,         1290,          65,           1;
  4:      170389,       295210,       158551,        32220,        2675,          90,         1;
  5:     2032785,      4095687,      2743013,       749091,       90195,        4949,       119,        1;
  6:    26546673,     60838072,     48743948,     16920680,     2742166,      216776,      8428,      152,       1;
  7:   376085653,    964040905,    897557556,    380270164,    78841686,     8363838,    465444,    13476,     189,     1;
  8:  5736591885,  16232914246,  17195531101,   8623617320,  2203222330,   299108292,  22239042,   916200,   20505,   230,   1;
  9: 93614616409, 289397068907, 343258226399, 199048582821, 60873452090, 10214666638, 969694110, 53129802, 1683165, 29975, 275, 1;
%F Formula
The row polynomials satisfy P(n,x) = (5+x)*P(n-1,x) + (4+4*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 29 + 14*x + x^2.
%t Mathematica
P[0, x_] := 29 + 14*x + x^2;
P[n_, x_] := P[n, x] = Expand[(5 + x)*P[n - 1, x] + (4 + 4*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] = 29 + 14*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (5 + x)*v[i - 1] + (4 + 4*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.