Real-rooted atlas

A038320 → x ↦ x+1 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(6 + 17\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 36 + 204\,x + 289\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
036204289529
121618365202491312167
21296146886242411791283521279841
377761101606242401768680250563014198576436343
446656793152561816021224160451013405111485224137569148035889
5279936555206447192544222853680631418760107341189210137778984103386733404825447
616796163807129637754035221393953287577025120171745902722433066955219696256304697575744178310985281
710077696256981248291245414419254557952818318712962318569686724379520519365317989202083766909018141185878764971801152661463
8604661761713208320218434060801650390681608183187129602782283624064656928077904010635978404160113007270544207115272589820201599390044941426511213649
93627970561130717491216018497792013615723123207715576436480306051198647048671450628332817549364366864024861599519724023480399546406013305559742963434271896307633952809757913927
102176782336740105994241153331841024108925784985606944018792832031479551860838410405740753999362527108468828416447508791355032056352958911374404790001507466824246757653414957658262223722976121914624432020321
1113060694016481068896256817817123635284962112288768601814962045440306925630643174411594968268742144328524100947694086981137145138499210988826987718008012454003919413742496235484831833464454445345039213589904578032905937504036361936467383
1278364164096310844517580857247198654464648801584750592505524568118169628646392193362944121747166821792512394228921137232896977359200319389888184612293393662534426153408230768859042694593575291336992190867044916469703683198455476409870816837782655940092911592836324538749809
134701849845761998286184448039632675991552048660118856294404136110102785024025781752974026649612174716682179251204435075362793870080125660468612492985602769184400904938016047076134815383946272606283554440550823205726011347494091108037439304964384441860151540043903460836102862423051509815793266635235464391245607
14282110990745612789031580467227176692108492803593362623234048033088880822280192022500438959150530561168772801489208115247307470536467947520150792562334991582720379773860695534356480753218157046143140352116406442452585758054413742427233985818659201198057758860302139520727392210736612013280274792612944942316128486611918756668684816132610415680998648961
Technical details
Node id
n-c4b1d536a6354c0e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-dde9c6f2b382eac4

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:        36,         204,          289;
  1:       216,        1836,         5202,          4913;
  2:      1296,       14688,        62424,        117912,         83521;
  3:      7776,      110160,       624240,       1768680,       2505630,        1419857;
  4:     46656,      793152,      5618160,      21224160,      45101340,       51114852,       24137569;
  5:    279936,     5552064,     47192544,     222853680,     631418760,     1073411892,     1013777898,       410338673;
  6:   1679616,    38071296,    377540352,    2139395328,    7577025120,    17174590272,    24330669552,     19696256304,      6975757441;
  7:  10077696,   256981248,   2912454144,   19254557952,   81831871296,   231856968672,   437952051936,    531798920208,    376690901814,    118587876497;
  8:  60466176,  1713208320,  21843406080,  165039068160,  818318712960,  2782283624064,  6569280779040,  10635978404160,  11300727054420,   7115272589820,   2015993900449;
  9: 362797056, 11307174912, 160184977920, 1361572312320, 7715576436480, 30605119864704, 86714506283328, 175493643668640, 248615995197240, 234803995464060, 133055597429634, 34271896307633;
%F Formula
The row polynomials satisfy P(n,x) = (6+17*x)*P(n-1,x) for n >= 1, with P(0,x) = 36 + 204*x + 289*x^2.
%t Mathematica
P[0, x_] := 36 + 204*x + 289*x^2;
P[n_, x_] := P[n, x] = Expand[(6 + 17*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] = 36 + 204*x + 289*x^2; for(n = 1, N, my(i = n + 1); v[i] = (6 + 17*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.