Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - 8\,x + 4\,x^{2} + 12\,n x - 16\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(1 + 8\,x + 24\,x^{2} + 32\,x^{3} + 16\,x^{4} + 12\,n x + 64\,n x^{2} + 112\,n x^{3} + 64\,n x^{4} + 16\,n^{2} x^{2} + 64\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
11163653
21322165124001161
3148540256060006912313619197
4164100871682800064512878086553620736274833
5180162015360840002903046585609830409331205120001239043601989
61962376281601980009123842897664648806410264320112640008177664353894469222444463897
7111232764659240040023063049417408281149446227020810250240012402790410734796862992384224788483686400525595149
811284320716807280005031936251130889371648026687232058572800099222323212881756161259847680899153920442368000134217728189399046012192033
9114455081044481224000987033658216704260702208907365888248934400054217912329385279488128504463361375705497611280384000684510412828978053127644119049463398467023744597
1011606840145920193800017860608121551360635043840261211392085995520002289200742449533419520871994572801244685926401428848640001300569784329176383488048412753920179804569604194304000462422016731837303721
111176831619712029260003033676823398636813970964486630750720254679040008012202598420804036198444762388070479867346944011788001280001430626762752141316305715211183346155526922475929603229614080001068194856962233047449622187868167855725255453
121192993625907242504004896460842209305628352839681525072665666944204800243007463424736142819328187188168294440078886830087229974118400109681385144321392975013478414698112090112127373557104648913734860800491369634201620544036536326123851612161159641169921048576000083128426904241
1312081170033280059800007577856072200128053886976003239533440015997696000065814521344022785372979206685291724800166995361792003560214528000064811727585280100603750973440132691289702400147844307353600137950658560000106463420743680667681187430403317086289920012562779340800340787200000058975269683248922361856869035683752037
14122413608419328819000011321856011814566409699655680644496652803536332800001626005821440633299337216021058668933120601183302451201478858342400003140876029132805761851192115209119510455910401241892181770240144848191488000014372561800396801201826137374720835905745059840474873059082240214695936000000743088398008321849265278156829463475650562257542184968990579434799673
Technical details
Node id
n-7a31a72b212e6b04 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-38ec58a5b1e47e30

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: 1,  16,   36;
  2: 1,  32,  216,    512,     400;
  3: 1,  48,  540,   2560,    6000,    6912,     3136;
  4: 1,  64, 1008,   7168,   28000,   64512,    87808,     65536,     20736;
  5: 1,  80, 1620,  15360,   84000,  290304,   658560,    983040,    933120,     512000,     123904;
  6: 1,  96, 2376,  28160,  198000,  912384,  2897664,   6488064,  10264320,   11264000,    8177664,    3538944,      692224;
  7: 1, 112, 3276,  46592,  400400, 2306304,  9417408,  28114944,  62270208,  102502400,  124027904,  107347968,    62992384,    22478848,     3686400;
  8: 1, 128, 4320,  71680,  728000, 5031936, 25113088,  93716480, 266872320,  585728000,  992223232, 1288175616,  1259847680,   899153920,   442368000,  134217728,   18939904;
  9: 1, 144, 5508, 104448, 1224000, 9870336, 58216704, 260702208, 907365888, 2489344000, 5421791232, 9385279488, 12850446336, 13757054976, 11280384000, 6845104128, 2897805312, 764411904, 94633984;
%F Formula
The row polynomials satisfy (1+(12*n-8)*x+(16*n^2-16*n+4)*x^2)*P(n,x) = (1+(12*n+8)*x+(16*n^2+64*n+24)*x^2+(64*n^2+112*n+32)*x^3+(64*n^2+64*n+16)*x^4)*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[Together[((1 + (12*n + 8)*x + (16*n^2 + 64*n + 24)*x^2 + (64*n^2 + 112*n + 32)*x^3 + (64*n^2 + 64*n + 16)*x^4)*P[n - 1, x])/(1 + (12*n - 8)*x + (16*n^2 - 16*n + 4)*x^2)]];
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] = ((1 + (12*n + 8)*x + (16*n^2 + 64*n + 24)*x^2 + (64*n^2 + 112*n + 32)*x^3 + (64*n^2 + 64*n + 16)*x^4)*v[i - 1])/(1 + (12*n - 8)*x + (16*n^2 - 16*n + 4)*x^2)); 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.