Real-rooted atlas

A145324 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-1 - 3\,n - 12\,n x - 3\,n^{2} - 24\,n^{2} x + 81\,n^{2} x^{2} - n^{3} - 12\,n^{3} x + 81\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(12\,n x^{2} + 24\,n^{2} x^{2} + 36\,n^{2} x^{3} + 12\,n^{3} x^{2} + 36\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(1 + 4\,n + 12\,n x + 2\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-12\,n x^{2} - 12\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 24\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
132427
212204486702
360159679381036819962
436013176101493273888243000631917
52520118152124407951649929404100629856022232403
62016011568961553531487765552244709100334768464180033840863989326
718144012347424202290210145273003254935901001147006267212556074384564350976036830786022
818144001430697602773008108242372304001162229492253282914098085470855470364992289044481928493100801710833243265
91995840017914953604016029798841450330016024145726164758632036914408194231735845562692833938380821076439876640714256704000086073604467795
1023950080024132453120614942106336733011041721650347697812200218324933707788620800407753498116031849322982413775962030965369443805772320002852027019072004664940439216518
113113510400348210005760994897608508813464773022931210668387825270005448248792785692187683270524770864427442897247872070775761988038824735034454194176004484485306901952012218932286668800271045784393243802
124358914560053606477030401698410931966722574863898443136231408390162267001361891491257160085519677854850530211575877604042922624317621881039598535644367701093700120004096455532308589824225245922420851712055927071320440320016811099876385914301
136538371840008773269200640030541976058873605130442676866521651608802857873130034423951805871309361606499647396256043953917539721538176512131411026045846386072231244378846267939200286923489942812857200238541954969758758912119416529958010871040272421095331280896001108774955711395244187
14104613949440001521413765222400577500967598457601065206636769632256118647509644314738008852082714129588321646788664839436273685418069089997285327438725173348796186771136312110005226196428640312001718203226726429500464019182491236784134478592145086614761685583916806668144724303161164800140727734195368320000077498784177140779889382
Technical details
Node id
n-8afe893223937754 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4287c3b0f10678a7

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        3,         24;
  2:       12,        204,         486;
  3:       60,       1596,        7938,        10368;
  4:      360,      13176,      101493,       273888,        243000;
  5:     2520,     118152,     1244079,      5164992,       9404100,       6298560;
  6:    20160,    1156896,    15535314,     87765552,     244709100,     334768464,      180033840;
  7:   181440,   12347424,   202290210,   1452730032,    5493590100,   11470062672,    12556074384,     5643509760;
  8:  1814400,  143069760,  2773008108,  24237230400,  116222949225,  328291409808,   547085547036,   499228904448,   192849310080;
  9: 19958400, 1791495360, 40160297988, 414503300160, 2414572616475, 8632036914408, 19423173584556, 26928339383808, 21076439876640, 7142567040000;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3+3*n^2+3*n+1+(12*n^3+24*n^2+12*n)*x-(81*n^3+81*n^2)*x^2)*P(n-2,x) + ((12*n^3+24*n^2+12*n)*x^2+(36*n^3+36*n^2)*x^3)*d/dx P(n-2,x) + (2*n^2+4*n+1+(12*n^2+12*n)*x)*P(n-1,x) - (12*n^2+12*n)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 24*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 24*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + 3*n^2 + 3*n + 1 + (12*n^3 + 24*n^2 + 12*n)*x - (81*n^3 + 81*n^2)*x^2)*P[n - 2, x] + ((12*n^3 + 24*n^2 + 12*n)*x^2 + (36*n^3 + 36*n^2)*x^3)*D[P[n - 2, x], x] + (2*n^2 + 4*n + 1 + (12*n^2 + 12*n)*x)*P[n - 1, x] - (12*n^2 + 12*n)*x^2*D[P[n - 1, x], x])/(n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 3 + 24*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 + 3*n^2 + 3*n + 1 + (12*n^3 + 24*n^2 + 12*n)*x - (81*n^3 + 81*n^2)*x^2)*v[i - 2] + ((12*n^3 + 24*n^2 + 12*n)*x^2 + (36*n^3 + 36*n^2)*x^3)*deriv(v[i - 2], x) + (2*n^2 + 4*n + 1 + (12*n^2 + 12*n)*x)*v[i - 1] - (12*n^2 + 12*n)*x^2*deriv(v[i - 1], x))/(n)); 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. [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.