Real-rooted atlas

A038330 → basis A007318 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(16 + 20\,x + 4\,x^{2} + 12\,n x + 4\,n x^{2} + n^{2} x^{2}\right)\,P_{n}(x) = \left(256 + 576\,x + 272\,x^{2} + 36\,x^{3} + 192\,n x + 144\,n x^{2} + 24\,n x^{3} + 16\,n^{2} x^{2} + 4\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 512\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0256512144912
14096122886912102424320
265536262144221184655366400620800
310485765242880589824026214405120003686415360000
41677721610066329614155776083886080245760003538944200704371200000
52684354561879048192317089382423488102409175040001981808642247884810485768806400000
6429496729634359738368676457349126012954214429360128000845571686414386462721342177285308416205824000000
76871947673661847529062413915694039041443109011456845571686400304405807104690550210569663676416764411904262144004751360000000

All 15 rows

Technical details
Node id
n-479804acd890a484 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9e3a89ee8f65d0f5

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:            256,             512,             144;
  1:           4096,           12288,            6912,            1024;
  2:          65536,          262144,          221184,           65536,            6400;
  3:        1048576,         5242880,         5898240,         2621440,          512000,           36864;
  4:       16777216,       100663296,       141557760,        83886080,        24576000,         3538944,         200704;
  5:      268435456,      1879048192,      3170893824,      2348810240,       917504000,       198180864,       22478848,        1048576;
  6:     4294967296,     34359738368,     67645734912,     60129542144,     29360128000,      8455716864,     1438646272,      134217728,       5308416;
  7:    68719476736,    618475290624,   1391569403904,   1443109011456,    845571686400,    304405807104,    69055021056,     9663676416,     764411904,     26214400;
  8:  1099511627776,  10995116277760,  27831388078080,  32985348833280,  22548578304000,   9740985827328,  2762200842240,   515396075520,   61152952320,   4194304000,   126877696;
  9: 17592186044416, 193514046488576, 544258255749120, 725677674332160, 566935683072000, 285735584268288, 97229469646848, 22677427322880, 3587639869440, 369098752000, 22330474496, 603979776;
%F Formula
The row polynomials satisfy (16+(12*n+20)*x+(n^2+4*n+4)*x^2)*P(n,x) = (256+(192*n+576)*x+(16*n^2+144*n+272)*x^2+(4*n^2+24*n+36)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 512*x + 144*x^2.
%t Mathematica
P[0, x_] := 256 + 512*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((256 + (192*n + 576)*x + (16*n^2 + 144*n + 272)*x^2 + (4*n^2 + 24*n + 36)*x^3)*P[n - 1, x])/(16 + (12*n + 20)*x + (n^2 + 4*n + 4)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 256 + 512*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((256 + (192*n + 576)*x + (16*n^2 + 144*n + 272)*x^2 + (4*n^2 + 24*n + 36)*x^3)*v[i - 1])/(16 + (12*n + 20)*x + (n^2 + 4*n + 4)*x^2)); 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.