Real-rooted atlas

Veronese parent of A098172 → basis A013611

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(2 + 20\,x + 64\,x^{2} + 64\,x^{3}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1 + 8\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23Pn(1)
1181625
2236256896153610243750
34112136093443968010649617612816384065536562500
483045216533123604481691648561971213205504214958082306867214680064419430484375000
51676817024230912214041614338048715530242701393927750287361680867328271161753631541166082499805184120795955226843545612656250000
63218565043285248010037760873523205816320003025960960124498739204073062400010593134182421771793203234920097382442781900800038654705664024266565222494489280512171798691841898437500000
7644352140032283443240471552433246208360728576023917527040128233963520561344675840201692859596859576145674241443912640102428553211019264456122842480645794555127398457161719742464421937587159042192151307878471468255805441099511627776284765625000000
81289984371200875212814687232018662891521865107865615029870592099341251379254489566412802499844610457696386833973248313040033546240856259712516096196768021086208037796463824076806019826162073600785532338569216082545835455283206811474534072320424851292972646418823639067525125277655813324807036874417766442714843750000000

All 15 rows

Technical details
Node id
n-22c2c81430f30e6b · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-5719ac3c88c372d2

Observed on the stored rows ?

  • Degrees: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44 (affine: slope 3, 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1,     8,      16;
   2:   2,    36,     256,      896,       1536,        1024;
   3:   4,   112,    1360,     9344,      39680,      106496,       176128,        163840,          65536;
   4:   8,   304,    5216,    53312,     360448,     1691648,      5619712,      13205504,       21495808,        23068672,         14680064,          4194304;
   5:  16,   768,   17024,   230912,    2140416,    14338048,     71553024,     270139392,      775028736,      1680867328,       2711617536,       3154116608,        2499805184,         1207959552,          268435456;
   6:  32,  1856,   50432,   852480,   10037760,    87352320,    581632000,    3025960960,    12449873920,     40730624000,     105931341824,     217717932032,      349200973824,       427819008000,       386547056640,        242665652224,         94489280512,          17179869184;
   7:  64,  4352,  140032,  2834432,   40471552,   433246208,   3607285760,   23917527040,   128233963520,    561344675840,    2016928595968,    5957614567424,    14439126401024,     28553211019264,     45612284248064,      57945551273984,      57161719742464,       42193758715904,       21921513078784,        7146825580544,         1099511627776;
   8: 128,  9984,  371200,  8752128,  146872320,  1866289152,  18651078656,  150298705920,   993412513792,   5448956641280,   24998446104576,   96386833973248,   313040033546240,    856259712516096,   1967680210862080,    3779646382407680,    6019826162073600,     7855323385692160,     8254583545528320,     6811474534072320,      4248512929726464,      1882363906752512,       527765581332480,        70368744177664;
   9: 256, 22528,  950272, 25575424,  493182976,  7253917696,  84587905024,  802461319168,  6305910685696,  41578949771264,  232173543096320, 1105054215962624,  4502450522292224,  15741978020937728,  47263874193293312,  121748080730046464,  268364735068504064,   503936071982120960,   800781877651701760,  1066724191035719680,   1175760560139010048,   1052962703502475264,    746542006980837376,    403072166649659392,    155655662120992768,    38280596832649216,     4503599627370496;
  10: 512, 50176, 2367488, 71614464, 1560133632, 26069139456, 347454701568, 3792495181824, 34538924408832, 266047263604736, 1750861889929216, 9918210363097088, 48626144907362304, 207115543067492352, 768247612053651456, 2484417072092676096, 7004065626448723968, 17191931960428068864, 36647505406053056512, 67576337586345476096, 107187953717557067776, 145141525005780123648, 166071362159193882624, 158375273345971126272, 123541056128167182336, 76764981548436946944, 36533200177229463552, 12501992565580496896, 2738188573441261568, 288230376151711744;
%F Formula
The row polynomials satisfy P(n,x) = (2+20*x+64*x^2+64*x^3)*P(n-1,x) for n >= 2, with P(1,x) = 1 + 8*x + 16*x^2.
%t Mathematica
P[1, x_] := 1 + 8*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(2 + 20*x + 64*x^2 + 64*x^3)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 16*x^2; for(n = 2, N, my(i = n); v[i] = (2 + 20*x + 64*x^2 + 64*x^3)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

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.