Real-rooted atlas

A038319 → basis A013611 → x ↦ x+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(36 + 20\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1296 + 1440\,x + 400\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0129614404003136
14665677760432008000175616
216796163732480311040011520001600009834496
360466176167961600186624000103680000288000003200000550731776
42176782336725594112010077696000746496000031104000006912000006400000030840979456
578364164096304749527040507915878400470292480000261273600000870912000001612800000012800000001727094849536
6282110990745612538266255360243799621632002708884684800018811699200000836075520000023224320000003686400000002560000000096717311574016
71015599566684165077997833420801128443962982400146279772979200012189981081600006772211712000002508226560000005971968000000082944000000005120000000005416169448144896

All 15 rows

Technical details
Node id
n-2090bc852fe1ed0c · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-dca555f310728b4d

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:               1296,               1440,                 400;
  1:              46656,              77760,               43200,                8000;
  2:            1679616,            3732480,             3110400,             1152000,              160000;
  3:           60466176,          167961600,           186624000,           103680000,            28800000,             3200000;
  4:         2176782336,         7255941120,         10077696000,          7464960000,          3110400000,           691200000,            64000000;
  5:        78364164096,       304749527040,        507915878400,        470292480000,        261273600000,         87091200000,         16128000000,         1280000000;
  6:      2821109907456,     12538266255360,      24379962163200,      27088846848000,      18811699200000,       8360755200000,       2322432000000,       368640000000,        25600000000;
  7:    101559956668416,    507799783342080,    1128443962982400,    1462797729792000,    1218998108160000,     677221171200000,     250822656000000,     59719680000000,      8294400000000,      512000000000;
  8:   3656158440062976,  20311991333683200,   50779978334208000,   75229597532160000,   73139886489600000,   48759924326400000,   22574039040000000,   7166361600000000,   1492992000000000,   184320000000000,   10240000000000;
  9: 131621703842267136, 804354856813854720, 2234319046705152000, 3723865077841920000, 4137627864268800000, 3218155005542400000, 1787863891968000000, 709469798400000000, 197074944000000000, 36495360000000000, 4055040000000000, 204800000000000;
%F Formula
The row polynomials satisfy P(n,x) = (36+20*x)*P(n-1,x) for n >= 1, with P(0,x) = 1296 + 1440*x + 400*x^2.
%t Mathematica
P[0, x_] := 1296 + 1440*x + 400*x^2;
P[n_, x_] := P[n, x] = Expand[(36 + 20*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] = 1296 + 1440*x + 400*x^2; for(n = 1, N, my(i = n + 1); v[i] = (36 + 20*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.