Real-rooted atlas

A200536 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 7\,x + 17\,x^{2} + 17\,x^{3} + 6\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
115612
2110376036144
3115933055585402161728
41201748602641516062644320129620736
51252801850798523525479106660060480324007776248832
613041134001891574550213445447300680940734400532656233280466562985984
7135567563538381189525699685196182541981106822900829029673029604408992163296027993635831808

All 15 rows

Technical details
Node id
n-597121562fed1ded · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-9f63f55d8b2eb3fe

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0: 1;
  1: 1,  5,   6;
  2: 1, 10,  37,    60,     36;
  3: 1, 15,  93,   305,    558,    540,     216;
  4: 1, 20, 174,   860,   2641,   5160,    6264,     4320,     1296;
  5: 1, 25, 280,  1850,   7985,  23525,   47910,    66600,    60480,     32400,      7776;
  6: 1, 30, 411,  3400,  18915,  74550,  213445,   447300,   680940,    734400,    532656,    233280,     46656;
  7: 1, 35, 567,  5635,  38381, 189525,  699685,  1961825,  4198110,   6822900,   8290296,   7302960,   4408992,    1632960,    279936;
  8: 1, 40, 748,  8680,  69958, 415240, 1877596,  6597400, 18205345,  39584400,  67593456,  89691840,  90665568,   67495680,  34898688,  11197440,   1679616;
  9: 1, 45, 954, 12660, 117846, 817110, 4373544, 18476820, 62457921, 170195525, 374747526, 665165520, 944685504, 1058974560, 916370496, 590664960, 267058944, 75582720, 10077696;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+7*x+17*x^2+17*x^3+6*x^4)*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[Together[((1 + 7*x + 17*x^2 + 17*x^3 + 6*x^4)*P[n - 1, x])/(1 + 2*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] = ((1 + 7*x + 17*x^2 + 17*x^3 + 6*x^4)*v[i - 1])/(1 + 2*x + 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.