Real-rooted atlas

A038320 → x ↦ x+1 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(23 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 529 + 552\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
05295521441225
112167190449936172842875
2279841584016457056158976207361500625
3643634316790460175204809141120238464024883252521875
41480358894634166966044565604204915201645401603433881629859841838265625
53404825447124350146761946350123216924783680883032192027642746884807434243583180864339296875
6783109852813268632429125968807044486228320394244061948083201695421808644422839500865930526724299816962251875390625
7180115266146384575864103481765061511724821487705360128168164650644488773807859712305175925555268238095155289006211072515978035278815638671875
84142651121364921613831937556050745518462088070602460468992064463116080384040359516154675217547615719424052315872952320102357142732801186749480960619173642242758547353515625
9952809757913927546829948020166814265129078786960223280281233187202329881195476736017018262645221376887909355402854433089789642342408632119037132801501238093414401566509314867274300837068896549157373046875
10219146244320203211372046051396054883937175625745200966847261957817740808038090124394739206710057842972999684084383034853130241826556388257300485956162135621632013811390459412480216178285451673620507031030988889161004482563379220508056640625
1150403636193646738334186814113951700761070195920088922806420473313253875044992267043216354891891202507884118811158630417446150391729799168910233933481554739235617849571017359361032401436841082880215457691166834688306580113913282562665914034028544106993205379072118272717781982421875
121159283632453874980984678108805326520344287169238557194286384599309715249796771584859879156662751889664897265206952436754432702207553267124416512418707609401515180032191149126031126495232664866525325657374721734434413893019238432906265560025661444292121594785955843445181213206118412839184645488644139545122369384765625
1326663523546439124560720867105384169749656207621029792479386830960172301543134316571830402696893718624085472128030955649639859068027904269179562085731026329601805676565544034213888094209212115340915507203822982520622529904640119675974558618327449628381654045522132992049359398340038492160594293759278055424044295187026935808015407021574586368144884079282928466796875
146132610415680998648961511939652091631191565442003242116880295966995204877459067186807571814408270474070447195447859201035607187951648821297152990580788475490176892928738321084578005100789760433362375730568211333120200979652512727286415360734012643959525741690882088889737750428988211245410646472835412787207290003447144146534408150314412956188672056697839394477834241848842588950364165070942774902496337890625
Technical details
Node id
n-30a6e10f806704ab · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4b5617cf7459dd0c

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:             529,              552,               144;
  1:           12167,            19044,              9936,              1728;
  2:          279841,           584016,            457056,            158976,             20736;
  3:         6436343,         16790460,          17520480,           9141120,           2384640,            248832;
  4:       148035889,        463416696,         604456560,         420491520,         164540160,          34338816,          2985984;
  5:      3404825447,      12435014676,       19463501232,       16924783680,        8830321920,        2764274688,        480743424,         35831808;
  6:     78310985281,     326863242912,      596880704448,      622832039424,      406194808320,      169542180864,      44228395008,       6593052672,       429981696;
  7:   1801152661463,    8457586410348,    17650615117248,    21487705360128,    16816465064448,     8773807859712,    3051759255552,     682380951552,     89006211072,      5159780352;
  8:  41426511213649,  216138319375560,   507455184620880,   706024604689920,   644631160803840,   403595161546752,  175476157194240,   52315872952320,  10235714273280,   1186749480960,    61917364224;
  9: 952809757913927, 5468299480201668, 14265129078786960, 22328028123318720, 23298811954767360, 17018262645221376, 8879093554028544, 3308978964234240, 863211903713280, 150123809341440, 15665093148672, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (23+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 529 + 552*x + 144*x^2.
%t Mathematica
P[0, x_] := 529 + 552*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(23 + 12*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] = 529 + 552*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (23 + 12*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. [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.