Real-rooted atlas

A038305 → basis A013610 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(169 + 585\,x + 324\,x^{2} + 351\,n x + 324\,n x^{2} + 81\,n^{2} x^{2}\right)\,P_{n}(x) = \left(2197 + 13689\,x + 17901\,x^{2} + 6561\,x^{3} + 4563\,n x + 9477\,n x^{2} + 4374\,n x^{3} + 1053\,n^{2} x^{2} + 729\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 169 + 936\,x + 729\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
01699367291834
1219718252284311166460544
2285613163687392066065281640251854688
33712935140980160161301971216010661625212576454027952
4482680980199288312314535512516160415803375165809592260406091517510368
5627485171216355868568412453711659742640126127023757544336436236969541930611001641455815808
681573072118071572896985248253082425226469123279302617502615369964481232241617883183544166434867844011107948421888
7106044993732642967536041646772080148472919161478476735681249507649957146104480574230973218623733373444079537749173874204890029089201689856
81378584918493817619774280267600463024058782784427456016626064270725019889888579870415618662506629080702844618240265169953696055036466357000421900912521752567729672704
917921603940375459196277220442518740236043515699227164077603396467415305250474042344486911244669374768958942885126695102080126397677928450536010734452550060331830490503451887258369619229384100990976
102329808512248177421329022398466329234768227862721199375106784066231114598452375105643722485654496116140374399293244900159528871848964929509439209569518725581915326000470588277825923470494412305657647730591665289486152826774224896
11302875106592253109035038373211081019058243257318944598826943930464961243673151904272375223172363750945122828039604676400797482535449339655707904166617419045283449179115583592252350026509806317527018259567778398280672806646999143384149820570235248412178362038014898176
12393737638569928915264905372249551215455716689402670597608968216321314752226348513646577572254513041133630223594463790100638811814267659216828310484055045054061711040264622728798072427579855400120619618744747932813613778556162360768734048769220479531906734378281520885147278302366225302632471940160643072
13511858930140907572126183248277616060231835750341040058851236457335152213647204012541832825693326258800430210578936009081382118847174255975785160684101900680488416014079171909326451448959359373538963453005004704165131045169379591761717046129150874404771316999933116951588406601882448285801003719268961413875527081298162301447467316768344916983808
1466541660918317984129483074376116276032344441686220973801720197833173624354183552069550725102312017655001664081348909471536262428748072021224524296328371358813281572684339203660584696424877376727027810709944348546072000163077724542899205159127328742911897267637504248108483996522081478061295243971830811488010438680397198704300010963291001775869952535522834578182049183095756434433902968832
Technical details
Node id
n-38e6a8342defa32a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-238a5270d7b73261

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:           169,            936,             729;
  1:          2197,          18252,           28431,            11664;
  2:         28561,         316368,          739206,           606528,           164025;
  3:        371293,        5140980,        16016130,         19712160,         10661625,          2125764;
  4:       4826809,       80199288,       312314535,        512516160,        415803375,        165809592,         26040609;
  5:      62748517,     1216355868,      5684124537,      11659742640,      12612702375,       7544336436,       2369695419,        306110016;
  6:     815730721,    18071572896,     98524825308,     242522646912,     327930261750,     261536996448,     123224161788,      31835441664,       3486784401;
  7:   10604499373,   264296753604,   1646772080148,    4729191614784,    7673568124950,    7649957146104,    4805742309732,    1862373337344,     407953774917,     38742048900;
  8:  137858491849,  3817619774280,  26760046302405,   87827844274560,  166260642707250,  198898885798704,  156186625066290,   80702844618240,   26516995369605,   5036466357000,   421900912521;
  9: 1792160394037, 54591962772204, 425187402360435, 1569922716407760, 3396467415305250, 4740423444869112, 4466937476895894, 2885126695102080, 1263976779284505, 360107344525500, 60331830490503, 4518872583696;
%F Formula
The row polynomials satisfy (169+(351*n+585)*x+(81*n^2+324*n+324)*x^2)*P(n,x) = (2197+(4563*n+13689)*x+(1053*n^2+9477*n+17901)*x^2+(729*n^2+4374*n+6561)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 169 + 936*x + 729*x^2.
%t Mathematica
P[0, x_] := 169 + 936*x + 729*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((2197 + (4563*n + 13689)*x + (1053*n^2 + 9477*n + 17901)*x^2 + (729*n^2 + 4374*n + 6561)*x^3)*P[n - 1, x])/(169 + (351*n + 585)*x + (81*n^2 + 324*n + 324)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 169 + 936*x + 729*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((2197 + (4563*n + 13689)*x + (1053*n^2 + 9477*n + 17901)*x^2 + (729*n^2 + 4374*n + 6561)*x^3)*v[i - 1])/(169 + (351*n + 585)*x + (81*n^2 + 324*n + 324)*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

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.