Real-rooted atlas

A370258 → reverse

nonnegative coefficients

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

Recurrence

\(\left(12 + 14\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(60 + 25\,x + 81\,n + 35\,n x + 27\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-15\,x - 13\,x^{2} - 9\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 15 + 10\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01510126
18484211190
24956602703611462
330035005286066055111584
41856437128273009100136578193536
51162802713202441881071002380025201051765314
673547119612562089164113954433915054264428413616323270
7468682514060475173052001130606442397749216901117206840171152638760
83004501510015005013986472510656360048348300133887602231460212520103952101440815036
919353672070963464011088041259657326255150574001740538803729726049335003795001518025313709445084
10125167770050067108008654457240848476200052065585002082623400548648100940539601013512564350021450300131340292076
118122425444351971769246668938785672697440816504843339002336703454873308343681555025472218675457195945751045044294843511265683004240
1252860229080246681069040508403666680610084400016473206489756249056047240908718010202307855264040387467124751466723599596016361803958540612258793820988
1334486742558417243371279203840569057640503144318404043126655863202545524565584106063523566031469397102066251362320974869896097486989663303240248248052080465119251776923210
14225484891364712025860872784287853052806004088347706520038385042355660251247549964321178331532778440079303836009897632959501759579192802210753344819039980961072071003665200673205281164440378882630
Technical details
Node id
n-b4b33b797f6b9147 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-d3523b7e079d5382

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        15,        10,          1;
  1:        84,        84,         21,         1;
  2:       495,       660,        270,        36,         1;
  3:      3003,      5005,       2860,       660,        55,         1;
  4:     18564,     37128,      27300,      9100,      1365,        78,        1;
  5:    116280,    271320,     244188,    107100,     23800,      2520,      105,       1;
  6:    735471,   1961256,    2089164,   1139544,    339150,     54264,     4284,     136,      1;
  7:   4686825,  14060475,   17305200,  11306064,   4239774,    921690,   111720,    6840,    171,     1;
  8:  30045015, 100150050,  139864725, 106563600,  48348300,  13388760,  2231460,  212520,  10395,   210,   1;
  9: 193536720, 709634640, 1108804125, 965732625, 515057400, 174053880, 37297260, 4933500, 379500, 15180, 253, 1;
%F Formula
The row polynomials satisfy (4*n^2+14*n+12)*P(n,x) = (27*n^2+81*n+60+(12*n^2+35*n+25)*x)*P(n-1,x) - ((9*n+15)*x+(8*n+13)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 15 + 10*x + x^2.
%t Mathematica
P[0, x_] := 15 + 10*x + x^2;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 + 81*n + 60 + (12*n^2 + 35*n + 25)*x)*P[n - 1, x] - ((9*n + 15)*x + (8*n + 13)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 14*n + 12)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 15 + 10*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = ((27*n^2 + 81*n + 60 + (12*n^2 + 35*n + 25)*x)*v[i - 1] - ((9*n + 15)*x + (8*n + 13)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 14*n + 12)); 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.