Real-rooted atlas

A272867 → basis A013611 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(46 + 56\,x + 16\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1465616118
2211651524608179225613924
3973363554885343364229121858564300840961643032
444774562180326446044160550645764078182419152896557056091750465536193877776
520596297612536876803410653440546029568056962867204045848576198131712066060288014352384018350080104857622877577568
694742968966920355993623039197593646222919680062237620224059246582169640884869529620607506841675296931840194510848003372220416352321536167772162699554153024
743581765721637139243832321462501899980835271750664192582004119388166950216227225661943145234432418544332308482154544182067284085791784962449133797376516268490752744572846086576668672268435456318547390056832

All 15 rows

Technical details
Node id
n-606569787e0702d3 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-915220761d896620

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:               1;
  1:              46,                56,                16;
  2:            2116,              5152,              4608,               1792,                256;
  3:           97336,            355488,            534336,             422912,             185856,              43008,               4096;
  4:         4477456,          21803264,          46044160,           55064576,           40781824,           19152896,            5570560,             917504,              65536;
  5:       205962976,        1253687680,        3410653440,         5460295680,         5696286720,         4045848576,         1981317120,          660602880,          143523840,           18350080,            1048576;
  6:      9474296896,       69203559936,      230391975936,       462229196800,       622376202240,       592465821696,       408848695296,       206075068416,        75296931840,        19451084800,         3372220416,          352321536,          16777216;
  7:    435817657216,     3713924383232,    14625018999808,     35271750664192,     58200411938816,     69502162272256,     61943145234432,     41854433230848,     21545441820672,      8408579178496,      2449133797376,       516268490752,       74457284608,       6576668672,        268435456;
  8:  20047612231936,   195246310432768,   887703721967616,   2500924384673792,   4886437290377216,   7020670543724544,   7672712359051264,   6506154658103296,   4326028908429312,   2263010315862016,    928267657805824,    295437110083584,    71522211332096,   12732430548992,    1571958030336,    120259084288,     4294967296;
  9: 922190162669056, 10103996564895744, 52088926390456320, 167877871092105216, 379031140450566144, 636606123427233792, 824285315610968064, 841285735079215104, 686105388386353152, 450454567931346944, 238645352482209792, 101781260954763264, 34686782410850304, 9317929401188352, 1929681561452544, 297280456359936, 32083405701120, 2164663517184, 68719476736;
%F Formula
The row polynomials satisfy P(n,x) = (46+56*x+16*x^2)*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[(46 + 56*x + 16*x^2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (46 + 56*x + 16*x^2)*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

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.