Real-rooted atlas

A034870 → basis A013610 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(4\,n + 12\,n x + 9\,n x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 12 + 18\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27Pn(1)
00
1121830
2964326483241500
35764320129601944014580437456250
4307232256145152362880544320489888244944524881875000
51536020736012441604354560979776014696640146966409447840354294059049058593750
67372812165129123840410572801231718402586608643879912964157049603117787201558893604676680863772921757812500
734406467092486038323233210777612454041603362591232672518246410087773696113487454089457287840567437270423213342885803335726696156651269531250
81572864353894403715891202415329280108689817603586763980889669099520172933263360259399895040302633210880272369889792185706743040928533715203214155168068874753606887475361464843750000
97077888180486144216583372816243752960852797030403325908418569977725255682351892381696440979821568066146973235207937636788224757683511603256826263370243278438271360140504497344042151349203279033779756697356880241198730468750
10314572808965324801210318848010287710208061726261248027776817561609721886146560270823971225606093539352576011171488813056016757233219584020565695314944020565695314944016610753908224010678341798144053391708990720200218908715205299912289520883318714920697356880201144409179687500
11138412032435997900865399685120621297008640419375480832021388149522432855525980897282749904938598407218500463820801564008433827840281521518089011242228227713351685278528464168960548154878971392046984703911833603288929273828352185002271652844881618649258608027206216419536064435775730480966536635957269038331139831471252441406250
1260397977620837302272343815487488361006261862427075469639680154330176946176694485796257792252991254351052875897376305315841897434407632896039846122560290816706363081750609921059544622625914881344806636409815041440864253296230401296777827966607369725833709749555260071325854335488300356629271677441185618273440832035568548203224967621831757833921039340694250086778308875544858306884765625000
1326172456969814671360017666408448002031636971520016761005015040010559433159475205279716579737600214959889317888007254896264478720020555539416023040049333294598455296010090901167865856001765907704376524800264886155656478720034056791441547264003746247058570199040351210661740956160027890258432370048001859350562158003200102753583698205440046239112664192448016513968808640160045038096750836800881180153820720011014751922759006608851153655423245811462402343750
141127428915245660871065689038698577921112983732224001001685359001600691162897711104038013959374110720171062817183498240641485564438118400203137095405404160054847015759459123201271453547151097856025429070943021957120440118535552303104006601778033284546560085823114432699105280965510037367864934409371126833276336128078092723610636134400554869351970309376003329216111821856256016646080559109281280680976022872652416022205739876282144005551434969070536009992582944326964811529903397300344640550188738908625848770141601562500
Technical details
Node id
n-b856b70338e77df7 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-484d946563c33157

Observed on the stored rows ?

  • Degrees: -1, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27 (affine: slope 2, intercept -1)
  • 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:       0;
  1:      12,        18;
  2:      96,       432,        648,         324;
  3:     576,      4320,      12960,       19440,       14580,         4374;
  4:    3072,     32256,     145152,      362880,      544320,       489888,       244944,         52488;
  5:   15360,    207360,    1244160,     4354560,     9797760,     14696640,     14696640,       9447840,       3542940,        590490;
  6:   73728,   1216512,    9123840,    41057280,   123171840,    258660864,    387991296,     415704960,     311778720,     155889360,      46766808,       6377292;
  7:  344064,   6709248,   60383232,   332107776,  1245404160,   3362591232,   6725182464,   10087773696,   11348745408,    9457287840,    5674372704,    2321334288,     580333572,      66961566;
  8: 1572864,  35389440,  371589120,  2415329280, 10868981760,  35867639808,  89669099520,  172933263360,  259399895040,  302633210880,  272369889792,  185706743040,   92853371520,   32141551680,    6887475360,    688747536;
  9: 7077888, 180486144, 2165833728, 16243752960, 85279703040, 332590841856, 997772525568, 2351892381696, 4409798215680, 6614697323520, 7937636788224, 7576835116032, 5682626337024, 3278438271360, 1405044973440, 421513492032, 79033779756, 6973568802;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (4*n+12*n*x+9*n*x^2)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 12 + 18*x.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 12 + 18*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n + 12*n*x + 9*n*x^2)*P[n - 1, x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 12 + 18*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n + 12*n*x + 9*n*x^2)*v[i - 1])/(n - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.