Real-rooted atlas

A200536 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 - 14\,x - 72\,x^{2} - 42\,x^{3} + 36\,x^{4} + 42\,n x + 96\,n x^{2} - 42\,n x^{3} - 144\,n x^{4} + 49\,n^{2} x^{2} + 168\,n^{2} x^{3} + 144\,n^{2} x^{4}\right)\,P_{n}(x) = \left(24 + 252\,x + 1098\,x^{2} + 2541\,x^{3} + 3294\,x^{4} + 2268\,x^{5} + 648\,x^{6} + 252\,n x + 2046\,n x^{2} + 6594\,n x^{3} + 10530\,n x^{4} + 8316\,n x^{5} + 2592\,n x^{6} + 294\,n^{2} x^{2} + 2037\,n^{2} x^{3} + 5274\,n^{2} x^{4} + 6048\,n^{2} x^{5} + 2592\,n^{2} x^{6}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1684162252
2361008591312096810027153
32169072889383659047411507348322857682225880
41296725769972726168960209000254164048048866328313528328503056158502825
577765443209622800794707203776827501131702516222077457028609459202338340400110224800022863772810349977500
6466563919104847156328752665605354869500214299056165899712508911429282995215614799462014770123200092255323248342843217925748065856637175615625
727993627433728701029728869649177665084682600325958983056115544068201829897208190085730042565926814897457640085052663221686339742504704319879864886497996019788813774950720037586164725000

All 15 rows

Technical details
Node id
n-ff998881405c954e · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d4629d03e97d7675

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        6,         84,         162;
  2:       36,       1008,        5913,        12096,          8100;
  3:      216,       9072,       88938,       365904,        741150,         734832,          285768;
  4:     1296,      72576,      997272,      6168960,      20900025,       41640480,        48866328,         31352832,          8503056;
  5:     7776,     544320,     9622800,     79470720,     377682750,     1131702516,      2220774570,       2860945920,       2338340400,       1102248000,         228637728;
  6:    46656,    3919104,    84715632,    875266560,    5354869500,    21429905616,     58997125089,     114292829952,     156147994620,     147701232000,       92255323248,       34284321792,        5748065856;
  7:   279936,   27433728,   701029728,   8696491776,   65084682600,   325958983056,   1155440682018,    2989720819008,    5730042565926,    8148974576400,     8505266322168,     6339742504704,     3198798648864,      979960197888,      137749507200;
  8:  1679616,  188116992,  5547771648,  80325955584,  710913218400,  4276122617088,  18545825959920,   60061099650048,  148221684433809,  281536404609600,   412171519803120,   461821242645504,   389267641867104,   239110288284672,   101108138284800,   26330359136256,    3184768606464;
  9: 10077696, 1269789696, 42447255552, 702616965120, 7178563310400, 50407952628096, 258581812243104, 1005888033690624, 3037474464607926, 7221221266562100, 13612385563613298, 20369232682235136, 24079771617658848, 22229907108990336, 15699517959844800, 8195324281159680, 2980943415650304, 674818305831936, 71607703822848;
%F Formula
The row polynomials satisfy (4+(42*n-14)*x+(49*n^2+96*n-72)*x^2+(168*n^2-42*n-42)*x^3+(144*n^2-144*n+36)*x^4)*P(n,x) = (24+(252*n+252)*x+(294*n^2+2046*n+1098)*x^2+(2037*n^2+6594*n+2541)*x^3+(5274*n^2+10530*n+3294)*x^4+(6048*n^2+8316*n+2268)*x^5+(2592*n^2+2592*n+648)*x^6)*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[((24 + (252*n + 252)*x + (294*n^2 + 2046*n + 1098)*x^2 + (2037*n^2 + 6594*n + 2541)*x^3 + (5274*n^2 + 10530*n + 3294)*x^4 + (6048*n^2 + 8316*n + 2268)*x^5 + (2592*n^2 + 2592*n + 648)*x^6)*P[n - 1, x])/(4 + (42*n - 14)*x + (49*n^2 + 96*n - 72)*x^2 + (168*n^2 - 42*n - 42)*x^3 + (144*n^2 - 144*n + 36)*x^4)]];
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] = ((24 + (252*n + 252)*x + (294*n^2 + 2046*n + 1098)*x^2 + (2037*n^2 + 6594*n + 2541)*x^3 + (5274*n^2 + 10530*n + 3294)*x^4 + (6048*n^2 + 8316*n + 2268)*x^5 + (2592*n^2 + 2592*n + 648)*x^6)*v[i - 1])/(4 + (42*n - 14)*x + (49*n^2 + 96*n - 72)*x^2 + (168*n^2 - 42*n - 42)*x^3 + (144*n^2 - 144*n + 36)*x^4)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.