Real-rooted atlas

A136124 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4 + 2\,x + n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
216281256
36419218860504
42561152190413683605040
510246400156801880011016252055440
64096337921139202006401944649820820160665280
7163841720327598081827840258361621430089645121814408648640
865536851968475955214909440286128643440819225288128103708801814400121080960
9262144412876828409856111992832278455296452374272479642624319652928121337280199584001816214400
1010485761966080016318464078888960024577351685150960640734706176070343232004321184256153588096023950080029059430400
11419430492274688908328960527695872020086505472525543997449635073536012364909568010873093862462318919168209244556803113510400494010316800
12167772164278190084925161472338244403201542234439684914286755841121164537856184350667776021660110940161771508817408956162691072305396421120435891456008892185702400
1367108864196293427226117931008209175183360112426038067242790663618561185608828518424191494778880363166445424643957620167987230397283025408155640260505604755302899200653837184000168951528345600
142684354568925478912135878672384125458762956878438444564483510443153817611588941493043228646339167846453233049463193673937111939891275480835897958454861263260876826804562840576078700195123200104613949440003379030566912000
Technical details
Node id
n-5d6891c3c4269a64 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d428b0220c07a2b5

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, 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:      4,       3;
  2:     16,      28,       12;
  3:     64,     192,      188,        60;
  4:    256,    1152,     1904,      1368,       360;
  5:   1024,    6400,    15680,     18800,     11016,      2520;
  6:   4096,   33792,   113920,    200640,    194464,     98208,     20160;
  7:  16384,  172032,   759808,   1827840,   2583616,   2143008,    964512,    181440;
  8:  65536,  851968,  4759552,  14909440,  28612864,  34408192,  25288128,  10370880,   1814400;
  9: 262144, 4128768, 28409856, 111992832, 278455296, 452374272, 479642624, 319652928, 121337280, 19958400;
%F Formula
The row polynomials satisfy P(n,x) = (4+(n+2)*x)*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[(4 + (n + 2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (4 + (n + 2)*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.