Real-rooted atlas

A038226 → basis A013609 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(16 + 11\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 256 + 352\,x + 121\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0256352121729
1409684485808133119683
2655361802241858568518414641531441
31048576360448049561603407360117128016105114348907
4167772166920601611894784010903552056221440154608961771561387420489
526843545612918456322664431616305299456020989337608658101761984148321948717110460353203
642949672962362232012856841207808781566607366716588032036941234176126985492482494357888214358881282429536481
76871947673642520176230411693048463361875759857664193437735321613298844303366095303639041795937679363086767886423579476917625597484987
81099511627776755914244096023386096926720428745110323205158339608576042556301770752243812145561609578334289920246941430912037727163056025937424601205891132094649
917592186044416133040906960896457328117678080943239242711040129695395872768012483181852753928582187523768324214467087564801448723061350403319990348928045649867297762853116706115559060566555523
1028147497671065623221685578629128780699859419136201224371778355203112689500946432034239584510410752274630000760586241618355361624883269538706944819202124793823313920438238726058496547798407573123138428376721150094635296999121
11450359962737049640250921669623808166035051887198208418546693298978816719377129107619840890229197270679552816043430831456256561029858696626176289281020890447872110489278812323840303845516733890565697103438760448652793102357968345227121439314052555153018976267
127205759403792793669355434261505638430993209685610332168523132663542841344161140476920106844162215681557651469107222849216063280775168179529554782920376321079982477991005388849499196907921080321701534893709787136425383723427446784731128274640924167733087520240544379749833583241109418989131512359209
1311529215046068469761188950301625810944057218233265742151680170462653270856826880351579222371142205440531763573836352585728609312428354154004480538588664348761128960370279706739773276160197996787631684321280816736748980697825282552302340564680704058490261971273932809279705024288652809113996005997784041772481694156512954312706550833698643
141844674407370955161620291418481080506777610462762654307136307203356803018256872898560750035674391770038272012375588627464205631488155983981658663425146881531985534147587211264011848950615672744837120724102537624445517824034847434623176440545281306778798369116520448374337676616153169920791868162072631705601166591488767716352010693755313704066564594972986357216179766443076872509863361
Technical details
Node id
n-68bcd9136aa2ea5a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-89a932e93fe55d02

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,             352,             121;
  1:           4096,            8448,            5808,            1331;
  2:          65536,          180224,          185856,           85184,            14641;
  3:        1048576,         3604480,         4956160,         3407360,          1171280,           161051;
  4:       16777216,        69206016,       118947840,       109035520,         56221440,         15460896,         1771561;
  5:      268435456,      1291845632,      2664431616,      3052994560,       2098933760,        865810176,       198414832,        19487171;
  6:     4294967296,     23622320128,     56841207808,     78156660736,      67165880320,      36941234176,     12698549248,      2494357888,       214358881;
  7:    68719476736,    425201762304,   1169304846336,   1875759857664,    1934377353216,    1329884430336,    609530363904,    179593767936,     30867678864,     2357947691;
  8:  1099511627776,   7559142440960,  23386096926720,  42874511032320,   51583396085760,   42556301770752,  24381214556160,   9578334289920,   2469414309120,   377271630560,   25937424601;
  9: 17592186044416, 133040906960896, 457328117678080, 943239242711040, 1296953958727680, 1248318185275392, 858218752376832, 421446708756480, 144872306135040, 33199903489280, 4564986729776, 285311670611;
%F Formula
The row polynomials satisfy P(n,x) = (16+11*x)*P(n-1,x) for n >= 1, with P(0,x) = 256 + 352*x + 121*x^2.
%t Mathematica
P[0, x_] := 256 + 352*x + 121*x^2;
P[n_, x_] := P[n, x] = Expand[(16 + 11*x)*P[n - 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 + 352*x + 121*x^2; for(n = 1, N, my(i = n + 1); v[i] = (16 + 11*x)*v[i - 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. [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.