Real-rooted atlas

A143491 → basis A013610 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(7 + 3\,x + n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 42 + 39\,x + 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04239990
133643818927990
23024495030158108111880
3302405857245000171453240243154440
433264073501267071632359587075123937292162160
539916809818064102536285895288201568540994145927218732432400
651891840139609872162751356107399628438897691137628818268741662126561518918400
7726485760211021372826973486001991848860936655650290937339597051007807590590490196838821612800

All 15 rows

Technical details
Node id
n-c411b12a37f3c590 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-083b14d21af5bf73

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           42,           39,            9;
  1:          336,          438,          189,           27;
  2:         3024,         4950,         3015,          810,           81;
  3:        30240,        58572,        45000,        17145,         3240,          243;
  4:       332640,       735012,       670716,       323595,        87075,        12393,         729;
  5:      3991680,      9818064,     10253628,      5895288,      2015685,       409941,       45927,        2187;
  6:     51891840,    139609872,    162751356,    107399628,     43889769,     11376288,     1826874,      166212,       6561;
  7:    726485760,   2110213728,   2697348600,   1991848860,    936655650,    290937339,    59705100,     7807590,     590490,     19683;
  8:  10897286400,  33832663200,  46790870184,  37969778700,  20025381330,   7174027035,  1768388517,   296229150,   32280120,   2066715,   59049;
  9: 174356582400, 574014470400, 850151912544, 747889069752, 434315437380, 174860576550, 49816297377, 10044831951, 1405169370, 129907800, 7144929, 177147;
%F Formula
The row polynomials satisfy P(n,x) = (n+7+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 42 + 39*x + 9*x^2.
%t Mathematica
P[0, x_] := 42 + 39*x + 9*x^2;
P[n_, x_] := P[n, x] = Expand[(n + 7 + 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 42 + 39*x + 9*x^2; for(n = 1, N, my(i = n + 1); v[i] = (n + 7 + 3*x)*v[i - 1]); 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

Same fingerprint ?

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.