Real-rooted atlas

A139548 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(32 + 96\,x + 72\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1329672200
2102461441382413824518440000
33276829491211059202211840248832014929923732488000000
410485761258291266060288198180864371589120445906944334430208143327232268738561600000000
5335544325033164803397386240135895449603567255552064210599936802632499206879707136038698352640128994508801934917632320000000000
61073741824193273528321594506608647972533043202690729902080645775176499211301065588736145299414712321362182012928090812134195204086546038784111451255603213931406950464000000000000
734359738368721554505728703515643084842210938585088174120121663488522360364990464117531082122854420148185506775042644449347764224264444934776422419833370108231681081820187721728405682570395648936190547066881003061300428812800000000000000

All 15 rows

Technical details
Node id
n-08ea8d2f5c08b286 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-6b852c5cd7d4c499

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:             32,              96,                72;
  2:           1024,            6144,             13824,             13824,               5184;
  3:          32768,          294912,           1105920,           2211840,            2488320,             1492992,              373248;
  4:        1048576,        12582912,          66060288,         198180864,          371589120,           445906944,           334430208,            143327232,             26873856;
  5:       33554432,       503316480,        3397386240,       13589544960,        35672555520,         64210599936,         80263249920,          68797071360,          38698352640,          12899450880,           1934917632;
  6:     1073741824,     19327352832,      159450660864,      797253304320,      2690729902080,       6457751764992,      11301065588736,       14529941471232,       13621820129280,        9081213419520,        4086546038784,        1114512556032,         139314069504;
  7:    34359738368,    721554505728,     7035156430848,    42210938585088,    174120121663488,     522360364990464,    1175310821228544,     2014818550677504,     2644449347764224,     2644449347764224,     1983337010823168,     1081820187721728,      405682570395648,       93619054706688,       10030613004288;
  8:  1099511627776,  26388279066624,   296868139499520,  2078076976496640,  10130625260421120,   36470250937516032,  100293190078169088,   214913978738933760,   362667339121950720,   483556452162600960,   507734274770731008,   415418952085143552,   259636845053214720,   119832390024560640,    38517553936465920,     7703510787293184,     722204136308736;
  9: 35184372088832, 949978046398464, 12112220091580416, 96897760732643328, 545049904121118720, 2289209597308698624, 7439931191253270528, 19131251634651267072, 39458206496468238336, 65763677494113730560, 88780964617053536256, 96851961400422039552, 84745466225369284608, 58669938156024889344, 31430324012156190720, 12572129604862476288, 3535911451367571456, 623984373770747904, 51998697814228992;
%F Formula
The row polynomials satisfy P(n,x) = (32+96*x+72*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[(32 + 96*x + 72*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] = (32 + 96*x + 72*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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.