Real-rooted atlas

A136572 → basis A013610 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(4\,n + 9\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
14913
232144162338
338425925832437413182
4614455296186624279936157464685464
5122880138240062208001399680015746400708588044555160
6294912039813120223948800671846400113374080010203667203826375203475302480
782575360130056192087787929603292047360074071065600999959385607499695392024106163760316252525680
826424115204756340736037456183296016855282483204740548198400853298675712095996101017606171177922560173564379072032890262670720
9951268147201926317998080173368619827209101852540928030718752325632069117192732672010367578909900809997308234547205623485881932801405871470483203848160732474240
10380507258880085614133248000866843099136000520105859481600020479168217088000552937541861376001036757890990080001332974431272960001124697176386560005623485881932800012652843234348800500260895221651200
1116742319390720041437240492032004661689555353600031466404498636800014159882024386560004460362837681766400100358163847839744001612899061840281600018145114445703168000136088358342773760006123976125424819200125263148020053120071537308016696121600
128036313307545600216980459303731200268513318388367360020138498879127552000101951150575583232000367024142072099635200963438372939261542400185805971924000440320026128964801812561920002612896480181256192000176370512412234792960072151573259550597120013528419986165736960011159820050604594969600
134178882919923712001222323254077685760016501363930048757760013613625242290225152007657664198788251648000310135400050924191744009304062001527725752320020934139503437382942720035326360412050583715840044157950515063229644800039742155463556906680320024387231761728101826560091452119106480381849600158282513838139122432001886009588552176549862400
1423401744351572787200737154947074542796800107808911009651884032009702801990868669562880060036087318499892920320027016239293324951814144009117980761497171237273600234462362438498688958464004615977760507942938869760069239666407619144083046400778946247085715370934272006373196567064943944007680035849230689740309685043200124093490849101071986688001994359674360552942643200343253745116496132074956800
Technical details
Node id
n-fe1d8780dd3ac346 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-12e1fe8c90bb4450

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:           4,             9;
  2:          32,           144,            162;
  3:         384,          2592,           5832,           4374;
  4:        6144,         55296,         186624,         279936,          157464;
  5:      122880,       1382400,        6220800,       13996800,        15746400,         7085880;
  6:     2949120,      39813120,      223948800,      671846400,      1133740800,      1020366720,        382637520;
  7:    82575360,    1300561920,     8778792960,    32920473600,     74071065600,     99995938560,      74996953920,     24106163760;
  8:  2642411520,   47563407360,   374561832960,  1685528248320,   4740548198400,   8532986757120,    9599610101760,   6171177922560,   1735643790720;
  9: 95126814720, 1926317998080, 17336861982720, 91018525409280, 307187523256320, 691171927326720, 1036757890990080, 999730823454720, 562348588193280, 140587147048320;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+9*n*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 + 9*n*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 + 9*n*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.