Real-rooted atlas

A109983 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(4 + 6\,x + 2\,x^{2} - n - n x\right)\,P_{n-1}(x) + \left(4 + 10\,x + 8\,x^{2} + 2\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
135210
2134455306148
363321675749462150202440
4321218464561082411251742030307007042256
51683143255457012248017930517883512302057610175703150252752800
6898991860428331120479022762053042360294911520884501072050388920946261386092413660480
74863958009732000151082014525049563419957655257003549906925361001901979649080998262397318485100600603432251113600
82657293622928230774809115467224986785650394176077345632092140424086090531563290172036482787616313959255470030138618482400684257400128704660568320
914625632243747716207381873236127223196581285468415120994266000014254786800163350193051507193925511222589402672688882832214386821215692478353533752764392201157013010939504862087140108800
1080974531380490201115004375567258363020385634710550043423041156097196001938124994402631827395352923124220602669828729052008581074501242415647006283614480025728652950839687706021329150704063659605462457046189001847561638884021248
1145046719844881345754512749142677893005171571082430521419345062124391576826023875731982083748960066005487000250643552717623179654774722189275362287476936022996570311001216071134850531785264010190133576490546562945501232310365020983662702536699881939938070543230970912737280
122515959695148375192503713418403135734804401394344531503471298851817212581636504646272081462340604850322463750572154296103824903326130161409570250949112086070844191775657758502609004266580322433023414666129140108102154832104162917330000132074246304033920093064068747799978105817151441162199220811246802704156587599919386624
1314099336193125830293333250908405022591147054321100903292758140966077422127120997201746564291092543660490580817360425731973678203779853138417426863859016795938963125401747186821076125156194832484509512009384318791607934580662303104493270989021102171252720822908893768414674030592456443120872246480047820242806844263725962494805232271444452649917323380195001040060011185644310405120
1479238482531891995145322174091940923160079947738628483810186151334463326047914411158182940432712955942082298634652685953471652112174605709310044193748523674856152649703346146311031302229201812205320535032094888802850738404651555522033800972362410147899314079344108603819055668200432239922741627018654624416227806865895439262862133683938740645523480951770611691496886524197092680475625446107960023624962900140408100040116600213540626285805568

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(n\right)\,P_{n}(x) = \left(1 + 2\,x + x^{2} - n - 2\,n x - n x^{2}\right)\,P_{n-2}(x) + \left(-3 - 5\,x - 2\,x^{2} + 6\,n + 10\,n x + 4\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 5\,x + 2\,x^{2}\)
Technical details
Node id
n-d9640fb05f96c50b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-fbebeadd27b9d988)
Fingerprint
f6-a7f447fa676c998a

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:       3,        5,         2;
  2:      13,       44,        55,        30,          6;
  3:      63,      321,       675,       749,        462,        150,         20;
  4:     321,     2184,      6456,     10824,      11251,       7420,       3030,         700,          70;
  5:    1683,    14325,     54570,    122480,     179305,     178835,     123020,       57610,       17570,        3150,         252;
  6:    8989,    91860,    428331,   1204790,    2276205,    3042360,    2949115,     2088450,     1072050,      388920,       94626,      13860,        924;
  7:   48639,   580097,   3200015,  10820145,   25049563,   41995765,   52570035,    49906925,    36100190,    19796490,     8099826,    2397318,     485100,      60060,      3432;
  8:  265729,  3622928,  23077480,  91154672,  249867856,  503941760,  773456320,   921404240,   860905315,   632901720,   364827876,  163139592,   55470030,   13861848,   2400684,   257400,    12870;
  9: 1462563, 22437477, 162073818, 732361272, 2319658128, 5468415120, 9942660000, 14254786800, 16335019305, 15071939255, 11222589402, 6726888828, 3221438682, 1215692478, 353533752, 76439220, 11570130, 1093950, 48620;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n-4+(n-6)*x-2*x^2)*P(n-1,x) + (4+10*x+8*x^2+2*x^3)*d/dx 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[(-(n - 4 + (n - 6)*x - 2*x^2)*P[n - 1, x] + (4 + 10*x + 8*x^2 + 2*x^3)*D[P[n - 1, x], x])/(n)]];
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] = (-(n - 4 + (n - 6)*x - 2*x^2)*v[i - 1] + (4 + 10*x + 8*x^2 + 2*x^3)*deriv(v[i - 1], x))/(n)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.