Real-rooted atlas

basis-map coefficients of A297191, shifted → basis A013610 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-5 + 3\,x + 6\,n\right)\,P_{n}(x) = \left(1 + 9\,x + 27\,x^{2} + 27\,x^{3} + 6\,n + 36\,n x + 54\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
1724940
213120378432811024
319288175554008505583272922528
42552847882419273710136080142884699846561458752
5318401012571280323190979776199017026244002066715787320590498912896
6371224184141663201002375423403212798324277136644222003543302600272806388503056531441167772160
743168030303334152251351113621608547296751651272483743509776304858567684046376448150803385279178928208847829693087007744
8492208464406048005454540360884161809727927005398402111001750495381744089844234481238044953612573894060892820880040176939609183300484304672155834574848
95528086747310134721065798083280960500726772236636899288999833782679564888064596653550124025574816187447743756218562551424190266506820117087081120461030381919298091736387420489996432412672
1061348094050160056019229805173303712121500972067814496003058009929011240252712033819936836483310108336016736405692502719022217120352236969036035594472660482711297722185147219785820051526925037092980917360348678440117592186044416
1167422412681924116403258832533275642426652509731715706748890214471710391628714400141280058719842488155251361065328010346622205348106480382362499282505383663989897661025054885487544124328390723684175140145517898826591800563813037641792051108186431381059609307863255777280
127350401664283497472525030665990513765396626620393603312962364458363191183837656672497983883839217687376846720531726766454521353084328774082907994261209845253150923475847914903735147759833103934489449907597146114367904121822938884816574235853382117175488287206062820716458890377451673922824295364815348024557502464
13795928213525491400081133650102301056010238097870834392988005637547451253198026917800153689793607356295863901392022069740306510066327663175200017091217493181003769306344442080708823601169898511298137096446200151314336113118751681978304583720015252485080202370110093125390934406098908935009150244772264950200064252719549427588118015382072254186582832992323792361095168
14856888268758672235212106867516717428001840111182016572771072012439288912057885013736600426198587314501978125013519207922010270487952745965255453280825068752997220021490162868764800484536015931681359428808958670556015760901249542449022477346433982440027093230076675262527248048534256264022462750754479854014803826584188096075175681872830175277571748453526806725776981758534854066918318544228767924549611585267068834414592
Technical details
Node id
n-3db5ec3dcca66950 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3fd060df7937e702

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:  1;
  1:  7,   24,     9;
  2: 13,  120,   378,     432,       81;
  3: 19,  288,  1755,    5400,     8505,     5832,       729;
  4: 25,  528,  4788,   24192,    73710,   136080,    142884,      69984,       6561;
  5: 31,  840, 10125,   71280,   323190,   979776,   1990170,    2624400,    2066715,      787320,       59049;
  6: 37, 1224, 18414,  166320,  1002375,  4234032,  12798324,   27713664,   42220035,    43302600,    27280638,      8503056,       531441;
  7: 43, 1680, 30303,  334152,  2513511, 13621608,  54729675,  165127248,  374350977,   630485856,   768404637,    644815080,    338527917,     89282088,      4782969;
  8: 49, 2208, 46440,  604800,  5454540, 36088416, 180972792,  700539840, 2111001750,  4953817440,  8984423448,  12380449536,  12573894060,   8928208800,   4017693960,    918330048,    43046721;
  9: 55, 2808, 67473, 1013472, 10657980, 83280960, 500726772, 2366368992, 8899983378, 26795648880, 64596653550, 124025574816, 187447743756, 218562551424, 190266506820, 117087081120, 46103038191, 9298091736, 387420489;
%F Formula
The row polynomials satisfy (6*n-5+3*x)*P(n,x) = (6*n+1+(36*n+9)*x+(54*n+27)*x^2+27*x^3)*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[((6*n + 1 + (36*n + 9)*x + (54*n + 27)*x^2 + 27*x^3)*P[n - 1, x])/(6*n - 5 + 3*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] = ((6*n + 1 + (36*n + 9)*x + (54*n + 27)*x^2 + 27*x^3)*v[i - 1])/(6*n - 5 + 3*x)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.