Real-rooted atlas

A034870 → basis A013609 → θ+1

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14161232
21612828825680768
364768288051204800230444816384
425640962150457344896008601650176163842304327680
5102420480138240491520107520015482881505280983040414720102400112646291456
6409698304811008360448010137600194641922649292825952256182476809011200297369658982453248117440512
71638445875244728322385510482001920196804608344408064449839104442810368328007680180404224715653121938227232112642457602147483648

All 15 rows

Technical details
Node id
n-af8dc2447e44dda7 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c94e3f620e523ccf

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      4,      16,        12;
  2:     16,     128,       288,       256,         80;
  3:     64,     768,      2880,      5120,       4800,        2304,         448;
  4:    256,    4096,     21504,     57344,      89600,       86016,       50176,       16384,         2304;
  5:   1024,   20480,    138240,    491520,    1075200,     1548288,     1505280,      983040,       414720,       102400,        11264;
  6:   4096,   98304,    811008,   3604480,   10137600,    19464192,    26492928,    25952256,     18247680,      9011200,      2973696,       589824,       53248;
  7:  16384,  458752,   4472832,  23855104,   82001920,   196804608,   344408064,   449839104,    442810368,    328007680,    180404224,     71565312,    19382272,     3211264,      245760;
  8:  65536, 2097152,  23592960, 146800640,  596377600,  1717567488,  3673686016,  5997854720,   7591034880,   7497318400,   5772935168,   3435134976,  1550581760,   513802240,   117964800,   16777216,   1114112;
  9: 262144, 9437184, 120324096, 855638016, 4010803200, 13476298752, 34065088512, 66739765248, 103238074368, 127454412800, 126179868672, 100109647872, 63263735808, 31444697088, 12032409600, 3422552064, 681836544, 84934656, 4980736;
%F Formula
The row polynomials satisfy (1+(2*n-1)*x)*P(n,x) = (4+(8*n+12)*x+(16*n+12)*x^2+(8*n+4)*x^3)*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[((4 + (8*n + 12)*x + (16*n + 12)*x^2 + (8*n + 4)*x^3)*P[n - 1, x])/(1 + (2*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 + (8*n + 12)*x + (16*n + 12)*x^2 + (8*n + 4)*x^3)*v[i - 1])/(1 + (2*n - 1)*x)); 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.