Real-rooted atlas

A038319 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(1 + x\right)\,P_{n}(x) = \left(16 + 42\,x + 26\,x^{2}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 832\,x + 676\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
02568326761764
1409619968324481757674088
265536425984103833611248644569763111696
31048576851968027688960449945603655808011881376130691232
4167772161635778566645350401439825920175478784011406120963089157765489031744
5268435456305345331214885584896403151257606551207936063874277376345985669128031810176230539333248
6429496729655834574848317559144448103206721945620963865395202725302501376221430828236810280717025282088270645769682651996416
7687194767361005022347264653264525721624769613266944603759323381769811089004953610628679755366474021162582016300710972989445429503678976406671383849472
81099511627776178670639513601306529051443205661625889587201610024862351360313954848158515242514719021465603947795337707520240568778391552086872058863616014116709565337617080198121677824
917592186044416314460325543936255499014504448012455576957091840404806251105484809209342212649779214965181095555891217370299485913088014113368332304384076447411799982080248454088349941763670344486987776717368321110468608
102814749767106565488762045857792490558107848540162657189750846259209715350026531635202525991006898225152478885795057788518466701950025906257926774416799506104320489263435519885312023851592481594408967047061415016529929542895666168217630129469486639681536
1145035996273704969513854212820172892760078574996684855269546817602191362245325339465089024065675766179353853952142297493388600016896231233426756475027456281815738859453939712254416986470340362240165371041205721235456732894387161719111681984922298562989260824811528732037365761265437718438866624512
127205759403792793616393102643628605441731521466733271449611254889533766264422450295287604017994137616345968471305848094723984329814880800473088739946965620720087859210521120917419613749248113978809938712482283529260778307520389185536547227809080750270054422231129743905479720965557782435976369930246450997470329715097653148384174432398229504
1311529215046068469762810246167479189504031966550155075780608022509779067532528844801097351729542210781184039230324331134035427328106248795063488012615680221984089686216026357760360724145740101042831360455915239754849929134080444517358760978680905728328336685448450162032640177849037951243837767680666933892317164391628801548239392879131623424016772593422857259253762232232135326160725639168
1418446744073709551616479615345916448342016584531202835671416832044326949548371749109760234101702302338299985920912996638979119369945088271996915362529312296140863142141066301447497318401154317266368323337060352016673471625320225979760640189660739738017570519777281681083829496064829607116811382338428879605617131520569116921443980280856576019817464228852884779827204293783916251458368962564360874289942887405977693753749683698750476845056
Technical details
Node id
n-c443a0508165a48c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-0e62401a26557dd3

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:            256,             832,              676;
  1:           4096,           19968,            32448,             17576;
  2:          65536,          425984,          1038336,           1124864,            456976;
  3:        1048576,         8519680,         27688960,          44994560,          36558080,          11881376;
  4:       16777216,       163577856,        664535040,        1439825920,        1754787840,        1140612096,          308915776;
  5:      268435456,      3053453312,      14885584896,       40315125760,       65512079360,       63874277376,        34598566912,         8031810176;
  6:     4294967296,     55834574848,     317559144448,     1032067219456,     2096386539520,     2725302501376,      2214308282368,      1028071702528,       208827064576;
  7:    68719476736,   1005022347264,    6532645257216,    24769613266944,    60375932338176,    98110890049536,    106286797553664,     74021162582016,     30071097298944,     5429503678976;
  8:  1099511627776,  17867063951360,  130652905144320,   566162588958720,  1610024862351360,  3139548481585152,   4251471902146560,   3947795337707520,   2405687783915520,   868720588636160,   141167095653376;
  9: 17592186044416, 314460325543936, 2554990145044480, 12455576957091840, 40480625110548480, 92093422126497792, 149651810955558912, 173702994859130880, 141133683323043840, 76447411799982080, 24845408834994176, 3670344486987776;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = (16+42*x+26*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 832*x + 676*x^2.
%t Mathematica
P[0, x_] := 256 + 832*x + 676*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((16 + 42*x + 26*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] = 256 + 832*x + 676*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((16 + 42*x + 26*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.