Real-rooted atlas

A200536 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(6 + 31\,x + 59\,x^{2} + 49\,x^{3} + 15\,x^{4}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28Pn(1)
011
16191540
2362285415702251600
321620528118171192029512825337564000
412961641690936287736568801719340568350256500506252560000
57776123120876960370044010243830194392992560957523127750137025004809375759375102400000
64665688646477176804071168014492574036677516467666158191693791090578587563612000030147187586568750113906254096000000
72799366205248638487364042029601758841560556491524413202583702238598249593300645925534780720275274818993751578917812562352281251514953125170859375163840000000
8167961642550272505191168373142246419190636640728705255042113315152484774817687528494141852811193704421880132082197030011386019610007496342437503643967250001233376875002597062500025628906256553600000000
91007769628721433638647964163262542105619361871417685781658614429403886256647973247284784173386114466943046332258289943346528616735498327955299004594357227600033508460396250189080775562507965190687500235888453125043825429687538443359375262144000000000
106046617619147622402879701632027349187328018395672313609315036402048368451276033601165741163884802996291964760206318122622563401098961472475901157953065564085018726824779751251821470568570000143926279700625090967152363750044911309359375016692619218750043940759765625730423828125057665039062510485760000000000
113627970561263743078421006957312022168159833601666570422528094944373907328425649965729472153922766882544045653603042876401123244005240962023092639764866166398346358447323195773159941216541570202750327560125713337547544943756012608081349375041567379465768750231797787859687501017193861406250033825927480468758013518261718751205199316406258649755859375419430400000000000
122176782336827177287681505970579456174817792512001452647726323209195668634746884607828461995264187468809704225286302223701935128017722490112830448042045260414929437684625457138699536814496372739958990412115636428467488420262782877593308985027691390801297575002461806133568409375183075009476782500011249581206043125005612590719450000002216564523808593756668769550781250014362054628906250197214433593750012974633789062516777216000000000000
1313060694016537665236992106401120583681347448824483841226332000258560853965854963020847297712767475456213823529552565504803441587484064672254197512469383912068353223016238205761572450042408313255231083449562567721878529307453475319428197770862390641930469598278127650519578450106801910962872196500992959033083530906257846109252774069062552203010144669312500288682328903048437501303048484745820312546780853281347656251285027336582031250253680039843750000320473454589843751946195068359375671088640000000000000
1478364164096347414460825674252222263296101869640235417610077746448946176765594325394196484644347690517050882309692598141879808959276227791526444833724593853648746048101361084991186836816262347753145758973056587796039957343158396114403752013522536087619381876484801385699012860093800338063402190367372524973339473997540991836429024839540003959417382468235813125329341736852038535625023419829780066563593751409724486170580937500708671217425087109375292051057965925781250961088795561425781252428761487851562500044257773317871093755176878881835937502919292602539062526843545600000000000000
Technical details
Node id
n-b946de2fe5de2258 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d7967bc76bb8e021

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:        6,        19,         15;
  2:       36,       228,        541,         570,          225;
  3:      216,      2052,       8118,       17119,        20295,        12825,          3375;
  4:     1296,     16416,      90936,      287736,       568801,       719340,        568350,        256500,          50625;
  5:     7776,    123120,     876960,     3700440,     10243830,     19439299,      25609575,      23127750,       13702500,        4809375,         759375;
  6:    46656,    886464,    7717680,    40711680,    144925740,    366775164,     676661581,     916937910,      905785875,      636120000,      301471875,       86568750,       11390625;
  7:   279936,   6205248,   63848736,   404202960,   1758841560,   5564915244,   13202583702,   23859824959,    33006459255,    34780720275,    27481899375,    15789178125,     6235228125,     1514953125,      170859375;
  8:  1679616,  42550272,  505191168,  3731422464,  19190636640,  72870525504,  211331515248,  477481768752,   849414185281,  1193704421880,  1320821970300,  1138601961000,   749634243750,   364396725000,   123337687500,   25970625000,    2562890625;
  9: 10077696, 287214336, 3864796416, 32625421056, 193618714176, 857816586144, 2940388625664, 7973247284784, 17338611446694, 30463322582899, 43346528616735, 49832795529900, 45943572276000, 33508460396250, 18908077556250, 7965190687500, 2358884531250, 438254296875, 38443359375;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (6+31*x+59*x^2+49*x^3+15*x^4)*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 + 31*x + 59*x^2 + 49*x^3 + 15*x^4)*P[n - 1, x])/(1 + 2*x + x^2)]];
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 + 31*x + 59*x^2 + 49*x^3 + 15*x^4)*v[i - 1])/(1 + 2*x + x^2)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.