Real-rooted atlas

A204420 → basis A013609 → x ↦ x+1

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1325
2454812105
3157521309001204725
49922515624086520201601680363825
59823275172538101137780035532005292003024042567525
6140472832526834068802006609220758419200153846000159667206652807023641625
7273922023375559787278050461057376780200063583720497188692007113506400544864320172972801552224799125
8698501159606251509634166040001343632494204006484793515200018680219157600330551020800035234559360020756736000518918400442384067750625
922561587455281875511360875057532504853208572732520025514233534389600823854058307280017028072485424002261949738048001868417591040087333966720017643225600157931112186973125
10900207339465746812521260639238096258000213074735304213783001202439843785980800042567185669528520009924846343251840001549584699927264001605039519283200010584612118080004022655436800067044257280069015896025707255625
1143480014496195571043751064697583457610627525011184456563193568554900670269833376398515260025610198022283394520006581522581910836560001165292956481445792001426059661783236480011853533634022080006388479665568000020133390461184002815858805760036233345413496309203125
122500100833531245335015625632201917156376073507000069208234122268028077290004368536543821195616270400178091050963443850191960049624445936237671699200097279348874472981216000135601906550682604032001337565628376558380800912600127935175680004096370597679360000108804784254566400129529505064960022500907501781208015140625
131687568062633590601135546875439236798248210076292303125049876567618312799319705750003294803337690647181368970000142054142119430576960925000042401053555132620906558000090475783458388098170400000140170961358947248329600001580866331907589927200000128478790054452277440000732805079310944640000027826175925580032000063145633719168000006476475253248000016313157938791375810976953125
1413213657930421014406891332031253530352805665698789830053000000414252311556792562793080391250028491644785506658593383145600001290203313030436365397967130000408709486081180933257248640000937391073676894961569645200001586107858115852684540928000019947415292219580539774400001859656745356459893043200001267571843394511951296000061359370032520175616000019970438120024071680000391697223316439040000349729663675392000013654113194768381553787709765625
Technical details
Node id
n-288a7b414c3511ae · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1840f5afd9477abc

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:                 1;
  1:                 3,                 2;
  2:                45,                48,                12;
  3:              1575,              2130,               900,               120;
  4:             99225,            156240,             86520,             20160,             1680;
  5:           9823275,          17253810,          11377800,           3553200,           529200,            30240;
  6:        1404728325,        2683406880,        2006609220,         758419200,        153846000,         15966720,          665280;
  7:      273922023375,      559787278050,      461057376780,      200063583720,      49718869200,       7113506400,       544864320,       17297280;
  8:    69850115960625,   150963416604000,   134363249420400,    64847935152000,   18680219157600,    3305510208000,    352345593600,    20756736000,    518918400;
  9: 22561587455281875, 51136087505753250, 48532085727325200, 25514233534389600, 8238540583072800, 1702807248542400, 226194973804800, 18684175910400, 873339667200, 17643225600;
%F Formula
The row polynomials satisfy P(n,x) = (4*n^2-1+(4*n-2)*x)*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[(4*n^2 - 1 + (4*n - 2)*x)*P[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*n^2 - 1 + (4*n - 2)*x)*v[i - 1]); 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.