Real-rooted atlas

A038337 → basis A013609 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(45 + 22\,x + 22\,n\right)\,P_{n}(x) = \left(1541 + 1980\,x + 484\,x^{2} + 506\,n + 484\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1541 + 1980\,x + 484\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
0154119804844005
1470811017066534010648224775
2135053741431284475064191664023425612119625
337218853148559070227872040164085680527076005153632635596875
499763316549128885969804898620100263697605414827440139148064011337990432661478125
5262023523531536355074103782924218925033181291603860152357601667766851523571466976024943578881652449359375
667756026395346116140141281351992465208022193155417664221459976830401358773629875248921160977928979688396805487587353682580949140625
71730672774710113415693226269445654978739867289231502703728010985611931743688769815082483844483952978588161383769981946882222472878208012072692177924086075469921875
84376800967355093807480104362220147572625488963403348031774256928049077326487050400483366924996721923215598863577408014092423646991360380536745035392054327114800640026559922791424200520370283203125
91097802547161698510591858340999317846070509262782862011904191789443047602025559223425441440237534260197323936019495799308201117441111678464265332480426295815321488640102322102553960960131471617817548805843183014113289772530498896484375
102734564005212970492898198724506884040139812530101190994964054204815124891856078567665810324114160106949526996863308032104515074486822531840735270145337870714883668538932075597184012504677249430333440270130350742456934431553188276211712012855002631049216473473995077197265625
11677161894949427918978208530549090956014414442417604484417720133287945363135415195228987564428142976770404494070484350539329952509792745351715101619242701873290330348723202628590769582887805696116578014952624532736035763376933370753638470233891193038802944752017653916379136028281005788308275222823285296682373046875
1216683603580097070377320856586364442379902121204411370456000722615642549421540727066885920102631717929614269700304178563396877360736905664230695618196661018937536224308807954754644712448164402212167771842584320899562174479477160171523590602860540835608268810013745541343811018752180266987395466260889617817033646634213376062218212734278205441095310836670089111328125
13409227122256217868257755055891814320332245090344133675013299268384980132485250750160079487716035103272771099830180884006773693383354541800220064982098682825484052981152010875650572128305178483840925273832813362909438848060280811128183008947584002968555175782274628566016107718085816225068248064027537800238695480301568045760081415772204687360419972935956377886720013688006801541205196852360822574526214599609375
149998821329914671710262514404794703418868962671049689836959320378475135840403783512015604474905043201165870206601408699491900802471270403085428044734233603973900118234664522795770884938553376493862666419507204785286251736454278532889603618848768336708268027494402121884551712041914954956809499376562503278811411251231597305172759353352765440745638898770831466627072011505391898822725749964809855364897109667741696030113614963390651432962494469587450428863525390625
Technical details
Node id
n-8ad9fc69d69d537e · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9f722f4116b0c75f

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:              1541,               1980,                484;
  1:             47081,             101706,              65340,               10648;
  2:           1350537,            4143128,            4475064,             1916640,              234256;
  3:          37218853,          148559070,          227872040,           164085680,            52707600,             5153632;
  4:         997633165,         4912888596,         9804898620,         10026369760,          5414827440,          1391480640,           113379904;
  5:       26202352353,       153635507410,       378292421892,        503318129160,        386015235760,        166776685152,         35714669760,          2494357888;
  6:      677560263953,      4611614014128,     13519924652080,      22193155417664,      22145997683040,      13587736298752,       4892116097792,        897968839680,        54875873536;
  7:    17306727747101,    134156932262694,    456549787398672,     892315027037280,    1098561193174368,     876981508248384,     448395297858816,     138376998194688,     22224728782080,      1207269217792;
  8:   437680096735509,   3807480104362220,  14757262548896340,   33480317742569280,   49077326487050400,   48336692499672192,   32155988635774080,   14092423646991360,   3805367450353920,    543271148006400,    26559922791424;
  9: 10978025471616985, 105918583409993178, 460705092627828620, 1190419178944304760, 2025559223425441440, 2375342601973239360, 1949579930820111744, 1111678464265332480, 426295815321488640, 102322102553960960, 13147161781754880, 584318301411328;
%F Formula
The row polynomials satisfy (22*n+45+22*x)*P(n,x) = (506*n+1541+(484*n+1980)*x+484*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1541 + 1980*x + 484*x^2.
%t Mathematica
P[0, x_] := 1541 + 1980*x + 484*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((506*n + 1541 + (484*n + 1980)*x + 484*x^2)*P[n - 1, x])/(22*n + 45 + 22*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1541 + 1980*x + 484*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((506*n + 1541 + (484*n + 1980)*x + 484*x^2)*v[i - 1])/(22*n + 45 + 22*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

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.