Real-rooted atlas

A038334 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(28 + 16\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 784 + 896\,x + 256\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
07848962561936
1219523763221504409685184
261465614049281204224458752655363748096
31721036849172480561971203211264091750401048576164916224
4481890304165219532823602790401798307840770703360176160768167772167256313856
51349292851253971714048925229383688811708416050352619520172637552643288334336268435456319277809664
63778019983361727094849536345418969907239476453703682819746693120128902705971236829344563260129542144429496729614048223625216
7105784559534085440348776038412435082916659216580110555545614211523333324881208704761856309366494330887576322310144108233175859268719476736618121839509504

All 15 rows

Technical details
Node id
n-021042e647efbf92 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-d36c4f1a2e9bd6b7

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:              784,               896,                256;
  1:            21952,             37632,              21504,               4096;
  2:           614656,           1404928,            1204224,             458752,              65536;
  3:         17210368,          49172480,           56197120,           32112640,            9175040,            1048576;
  4:        481890304,        1652195328,         2360279040,         1798307840,          770703360,          176160768,           16777216;
  5:      13492928512,       53971714048,        92522938368,        88117084160,        50352619520,        17263755264,         3288334336,         268435456;
  6:     377801998336,     1727094849536,      3454189699072,      3947645370368,      2819746693120,      1289027059712,       368293445632,       60129542144,        4294967296;
  7:   10578455953408,    54403487760384,    124350829166592,    165801105555456,    142115233333248,     81208704761856,     30936649433088,     7576322310144,     1082331758592,      68719476736;
  8:  296196766695424,  1692552952545280,   4352279020830720,   6632044222218240,   6632044222218240,   4547687466663936,   2165565460316160,   707123415613440,   151526446202880,   19241453486080,   1099511627776;
  9: 8293509467471872, 52130630938394624, 148944659823984640, 255333702555402240, 291809945777602560, 233447956622082048, 133398832355475456, 54448503002234880, 15556715143495680, 2963183836856320, 338649581355008, 17592186044416;
%F Formula
The row polynomials satisfy P(n,x) = (28+16*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 896*x + 256*x^2.
%t Mathematica
P[0, x_] := 784 + 896*x + 256*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 16*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] = 784 + 896*x + 256*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 16*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.