Real-rooted atlas

A038336 → x ↦ x/(x+1) → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(12 + 46\,x + 34\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 144 + 816\,x + 1156\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
014481611562116
1172814688416163930497336
220736235008998784188659213363364477456
3248832352512019975680565977608018016045435424205962976
429859845076172835956224013583462402886485760327135052815448044169474296896
5358318087106641926040645632285252710408082160128013739672217612976357094452523350144435817657216
64299816969746251776966503301125476852039681939718430720439669510963262286514053125042241613824178579390489620047612231936
75159780352131574398976149117652172898583336714244189791810355211871076796006422423145059123227228104714649619286574172876860716992766464922190162669056
8619173642241754325319680223676478259201690000057958408379583620710402849058431041536672694351773696010891241885859840115719445037260807286039131975680206437775405977642420747482776576
97430083706882315709421977632805883478016027885000956313601580150054191104062679285482913792177591308868255744359410982233374720509165558163947520480878582710394880272497863535890432701888436380323841951354384207722496
108916100448256303147415240704472404722083430444616001530101760284427009754398720128940244421994086442621914128381378561035103628832119193618329960093902110720230821719700989542401961984617458411110410107193483876663296238642068369310105689762301673555234816
11106993205379072394091639812915266995578768195584696009623869587456493006816907624448025143347662288846848949859800575356436482691269434963509903365718947549297458544649002047068338592153601020232001078373777408788361091742379737088372281626656123764736811383032455654359044129065876983540801536
121283918464548864509287657604382729379381027547381761062996516455369932882825145240480907264469342489696058474496199470558120824851660864590466439124237680641601305313803288392499230246878149617669636096428497440452916986511364414822113757326527692831271656639114396237824136312349452549932318722758702310349224820736189937030341242876870656
13154070215745863686547984169199206401298683526891175936015944947746830548992013553205584805966643208448164814529052540928398941116241649703321601453285494880295347814404117642235494170152140809074063444885300890828801542590785630501151440896198666995119079693746176018762993983468637742694401226811145072949390868480496566415862860467732480937958785518736439050248737103395697172336050176
14184884258895036416838141973657498419217810516940221841408023549461287626656972802168512893568954662912014745887676268891707801676596694318396743037747231003423890779634086707209882341365186008365137920248888597345425395862732804936290514017603684610867276288126125726602398531584900623711206494611649331207851591328466876101558272047670375922834604902318080180088086819597396297646083189059870763703892770816401906756202069927458308096
Technical details
Node id
n-cc7b9cf6ff870c45 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-dde9c6f2b382eac4

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:          144,            816,            1156;
  1:         1728,          14688,           41616,            39304;
  2:        20736,         235008,          998784,          1886592,           1336336;
  3:       248832,        3525120,        19975680,         56597760,          80180160,          45435424;
  4:      2985984,       50761728,       359562240,       1358346240,        2886485760,        3271350528,         1544804416;
  5:     35831808,      710664192,      6040645632,      28525271040,       80821601280,      137396722176,       129763570944,        52523350144;
  6:    429981696,     9746251776,     96650330112,     547685203968,     1939718430720,     4396695109632,      6228651405312,      5042241613824,      1785793904896;
  7:   5159780352,   131574398976,   1491176521728,    9858333671424,    41897918103552,   118710767960064,    224231450591232,    272281047146496,    192865741728768,     60716992766464;
  8:  61917364224,  1754325319680,  22367647825920,  169000005795840,   837958362071040,  2849058431041536,   6726943517736960,  10891241885859840,  11571944503726080,   7286039131975680,   2064377754059776;
  9: 743008370688, 23157094219776, 328058834780160, 2788500095631360, 15801500541911040, 62679285482913792, 177591308868255744, 359410982233374720, 509165558163947520, 480878582710394880, 272497863535890432, 70188843638032384;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (12+46*x+34*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 144 + 816*x + 1156*x^2.
%t Mathematica
P[0, x_] := 144 + 816*x + 1156*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((12 + 46*x + 34*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 144 + 816*x + 1156*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((12 + 46*x + 34*x^2)*v[i - 1])/(1 + 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

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.