Real-rooted atlas

A081578 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 4\,x - 6\,x^{2} - 5\,x^{3} + 2\,x^{4} + 24\,n x - 4\,n x^{2} - 16\,n x^{3} - 4\,n x^{4} + 44\,n^{2} x^{2} + 20\,n^{2} x^{3} - 6\,n^{2} x^{4} - 2\,n^{2} x^{5} + 24\,n^{3} x^{3} + 8\,n^{3} x^{4} - 2\,n^{3} x^{5} + 4\,n^{4} x^{4}\right)\,P_{n}(x) = \left(12 + 84\,x + 222\,x^{2} + 273\,x^{3} + 150\,x^{4} + 12\,x^{5} - 24\,x^{6} + 72\,n x + 396\,n x^{2} + 792\,n x^{3} + 648\,n x^{4} + 114\,n x^{5} - 60\,n x^{6} + 132\,n^{2} x^{2} + 540\,n^{2} x^{3} + 678\,n^{2} x^{4} + 228\,n^{2} x^{5} - 48\,n^{2} x^{6} + 72\,n^{3} x^{3} + 216\,n^{3} x^{4} + 138\,n^{3} x^{5} - 12\,n^{3} x^{6} + 12\,n^{4} x^{4} + 24\,n^{4} x^{5}\right)\,P_{n-2}(x) + \left(8 + 24\,x + 12\,x^{2} - 50\,x^{3} - 48\,x^{4} - 6\,x^{5} - 4\,x^{6} + 48\,n x + 128\,n x^{2} + 120\,n x^{3} + 12\,n x^{4} - 16\,n x^{5} - 12\,n x^{6} + 88\,n^{2} x^{2} + 200\,n^{2} x^{3} + 140\,n^{2} x^{4} + 10\,n^{2} x^{5} - 12\,n^{2} x^{6} + 48\,n^{3} x^{3} + 80\,n^{3} x^{4} + 28\,n^{3} x^{5} - 4\,n^{3} x^{6} + 8\,n^{4} x^{4} + 8\,n^{4} x^{5}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
12810
27563699
320240432128820
461976280821764006421
5182364014544193288800115247646
6547131286706812684899600311043136341431
716404592028598470528080320042854410035281922379112
84921157472115704035233285298400412416016307203031042073616219881
914762531432450057616326912306288003173299218289152567705687091251200108623794
10442871771480169895887156761616120560020982758416153536072818688184757762406400123904716766283
11132860584584062626896300529536791011200124375910412082882567279656962669967365703680064430082949124670930844
12398581191318882264438161219657600367574040067879296008000563200609664204829850508809174016001687572481680998469222430115219069
1311957426217858480578065648140576001635340880034727605632482616099844474564608027731206656113541632002991042560482181120429178881605632192374600134
14358722720088471228289768761856748659270212258800168675214464270495927744296550653952223976358144116044134400406257674249338978304133806899210757734436864001218969561375
Technical details
Node id
n-64b0c861287f610b · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-4b7f5e0d58f0c946

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:     2,      8;
  2:     7,     56,      36;
  3:    20,    240,     432,      128;
  4:    61,    976,    2808,     2176,      400;
  5:   182,   3640,   14544,    19328,     8800,     1152;
  6:   547,  13128,   67068,   126848,    99600,    31104,     3136;
  7:  1640,  45920,  285984,   705280,   803200,   428544,   100352,    8192;
  8:  4921, 157472, 1157040,  3523328,  5298400,  4124160,  1630720,  303104,  20736;
  9: 14762, 531432, 4500576, 16326912, 30628800, 31732992, 18289152, 5677056, 870912, 51200;
%F Formula
The row polynomials satisfy (4+(24*n-4)*x+(44*n^2-4*n-6)*x^2+(24*n^3+20*n^2-16*n-5)*x^3+(4*n^4+8*n^3-6*n^2-4*n+2)*x^4-(2*n^3+2*n^2)*x^5)*P(n,x) = (12+(72*n+84)*x+(132*n^2+396*n+222)*x^2+(72*n^3+540*n^2+792*n+273)*x^3+(12*n^4+216*n^3+678*n^2+648*n+150)*x^4+(24*n^4+138*n^3+228*n^2+114*n+12)*x^5-(12*n^3+48*n^2+60*n+24)*x^6)*P(n-2,x) + (8+(48*n+24)*x+(88*n^2+128*n+12)*x^2+(48*n^3+200*n^2+120*n-50)*x^3+(8*n^4+80*n^3+140*n^2+12*n-48)*x^4+(8*n^4+28*n^3+10*n^2-16*n-6)*x^5-(4*n^3+12*n^2+12*n+4)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 8*x;
P[n_, x_] := P[n, x] = Expand[Together[((12 + (72*n + 84)*x + (132*n^2 + 396*n + 222)*x^2 + (72*n^3 + 540*n^2 + 792*n + 273)*x^3 + (12*n^4 + 216*n^3 + 678*n^2 + 648*n + 150)*x^4 + (24*n^4 + 138*n^3 + 228*n^2 + 114*n + 12)*x^5 - (12*n^3 + 48*n^2 + 60*n + 24)*x^6)*P[n - 2, x] + (8 + (48*n + 24)*x + (88*n^2 + 128*n + 12)*x^2 + (48*n^3 + 200*n^2 + 120*n - 50)*x^3 + (8*n^4 + 80*n^3 + 140*n^2 + 12*n - 48)*x^4 + (8*n^4 + 28*n^3 + 10*n^2 - 16*n - 6)*x^5 - (4*n^3 + 12*n^2 + 12*n + 4)*x^6)*P[n - 1, x])/(4 + (24*n - 4)*x + (44*n^2 - 4*n - 6)*x^2 + (24*n^3 + 20*n^2 - 16*n - 5)*x^3 + (4*n^4 + 8*n^3 - 6*n^2 - 4*n + 2)*x^4 - (2*n^3 + 2*n^2)*x^5)]];
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] = 2 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = ((12 + (72*n + 84)*x + (132*n^2 + 396*n + 222)*x^2 + (72*n^3 + 540*n^2 + 792*n + 273)*x^3 + (12*n^4 + 216*n^3 + 678*n^2 + 648*n + 150)*x^4 + (24*n^4 + 138*n^3 + 228*n^2 + 114*n + 12)*x^5 - (12*n^3 + 48*n^2 + 60*n + 24)*x^6)*v[i - 2] + (8 + (48*n + 24)*x + (88*n^2 + 128*n + 12)*x^2 + (48*n^3 + 200*n^2 + 120*n - 50)*x^3 + (8*n^4 + 80*n^3 + 140*n^2 + 12*n - 48)*x^4 + (8*n^4 + 28*n^3 + 10*n^2 - 16*n - 6)*x^5 - (4*n^3 + 12*n^2 + 12*n + 4)*x^6)*v[i - 1])/(4 + (24*n - 4)*x + (44*n^2 - 4*n - 6)*x^2 + (24*n^3 + 20*n^2 - 16*n - 5)*x^3 + (4*n^4 + 8*n^3 - 6*n^2 - 4*n + 2)*x^4 - (2*n^3 + 2*n^2)*x^5)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.