Real-rooted atlas

A038284 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(32 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1024 + 1152\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0102411523242500
13276855296311045832125000
21048576235929619906567464961049766250000
3335544329437184010616832059719680167961601889568312500000
4107374182436238786565096079360382205952016124313603627970563401222415625000000
534359738368135291469824228304355328214035333120120394874880406332702727618738176612220032781250000000
610995116277764947802324992974098582732810958609055744770527199232034673723965449751984865281567283281921101996057639062500000000
735184372088832178120883699712400771988324352526013234675712443823666757632249650812551168936190547066882256887925964831737486458881983592903681953125000000000
811258999068426246333186975989760160308795329740802404631929946112023670595560407040159776520032747527489524376535040240734712102912050779978334208063474972917760357046722662497656250000000000
9360287970189639682229281815548395526269855106229862401058038049176289280119029280532332544093735558419211878452726251610806681621184654665056256059581841245470720111715952335257601256804463771648642684100792324882812500000000000
101152921504606846976778222015609621708824076243607922671616451429567648550092805713405465551962112051420649189967659008337448010309162762241626981478276320460857198567595651891201429964189891297280241306457044156416246790694704250881156831381426176244140625000000000000
1136893488147419103232269783632078002192384910519758263257399296187794700141796838604826408629707440180428802673873757878318268416200540531840873870131211280404916049155194884758920823958237347841487162757486949171203346116204345635635251332464498484183044812418546732892162082296486567116812207031250000000000000
1211805916207174113034249297159013149614014464339927376418282762403847648365969411362154086411831066108933200832102413309949372549850936115211230269783088936727347272194591462714593247232355332754855548388638721332497830708306457395237476501488671119114247665648031773638000641077981754468167843849328688259820683264374813367582081024610351562500000000000000
13377789318629571617095683187597375937010519244801255116466775197891952640305934638776454486163456051626470293526694540083206388775698823928449335296598947721764743292125184043316754877628755948339202436567461866617522094080106599826456664516591616035977441429124274349670491987776381283656007680172477080714906855014402238885182356963983360179910416439398891520674664061647745843230517578125000000000000000
14120892581961462917470617610880332376531662572355584459014022134929514771251201204911808104189976274534402202729399190472300376883202973684688907137605508792323066612335435485655680942082464242055260658116172185601559403175594635214140211207797015877973176070701056030700750019519380778385408941954830144344637518643222077066331508077441843203821030711222551864934404605706660848611622912034542799956364587171841214395310965942517761525878906250000000000000000
Technical details
Node id
n-d5e838559a8c8a11 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-f3c17ca3ebbc301e

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:              1024,               1152,                324;
  1:             32768,              55296,              31104,                5832;
  2:           1048576,            2359296,            1990656,              746496,              104976;
  3:          33554432,           94371840,          106168320,            59719680,            16796160,            1889568;
  4:        1073741824,         3623878656,         5096079360,          3822059520,          1612431360,          362797056,           34012224;
  5:       34359738368,       135291469824,       228304355328,        214035333120,        120394874880,        40633270272,         7618738176,          612220032;
  6:     1099511627776,      4947802324992,      9740985827328,      10958609055744,       7705271992320,      3467372396544,       975198486528,       156728328192,       11019960576;
  7:    35184372088832,    178120883699712,    400771988324352,     526013234675712,     443823666757632,    249650812551168,     93619054706688,     22568879259648,     3173748645888,      198359290368;
  8:  1125899906842624,   6333186975989760,  16030879532974080,   24046319299461120,   23670595560407040,  15977652003274752,   7489524376535040,   2407347121029120,   507799783342080,    63474972917760,    3570467226624;
  9: 36028797018963968, 222928181554839552, 626985510622986240, 1058038049176289280, 1190292805323325440, 937355584192118784, 527262516108066816, 211846546650562560, 59581841245470720, 11171595233525760, 1256804463771648, 64268410079232;
%F Formula
The row polynomials satisfy P(n,x) = (32+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 1024 + 1152*x + 324*x^2.
%t Mathematica
P[0, x_] := 1024 + 1152*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(32 + 18*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] = 1024 + 1152*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (32 + 18*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.