Real-rooted atlas

A038333 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(19 + 7\,x + 7\,n\right)\,P_{n}(x) = \left(312 + 266\,x + 49\,x^{2} + 84\,n + 49\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 312 + 266\,x + 49\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
031226649627
147526552279334314440
26912013305691728260682401322373
3974592241920023284801070160228095168077037334
413436928409328645080320032598720112366801915998117649151042039
51821450246584094721002855168829785600399334320110119464156473178235433199119908
624365629441020012134418435465216187199631361161699840044725443841027781664125178536576480167040380425
732248627200153503465472321303822336387144769536294839419392146374179840469617160329250034976985780971403536071392652169362
842310198886422574039040005372621291520749708918784067750334668804127751871488170769876480046961716032080937806040766718533028247524928719205102331
95510645415936325788531425288691005030400013789727981568014431896686592010433551538995252972815684096187846864128004520065168080692467896120590373270411977326743588582360748896
1071328803586048462894214938624136831183198617624334814085120002895842876129280242455864334745614606972154593286356737882091521972392073344004218727490208058167303274084508304974041384128720111998496666516557
1191835834617036864909211263303682106168677970739241505458903580672553617020436480005270434034555289636772472757436416189890638009712647230789340879104199430754082560038390420160892848120223617648341879793864796889010407243464641097277710
1211769252591697920899991179246960643180551351901880326880151014704414721016883743137726464108508936005550080086083755897736396851481461860410982423261603156189798478735261711794688195442139000908803420237432515904392981826210792257724767682626782230728494920199065084075903
1315021846035221708812357715221282816004724953691046543360111319297316565811201806039641359908864021354558605892255744189890638009712640001291256338466045952067569418691789414402713853701555476480826720247973844224186558405409958400299270775345141603174083980933320193293575761965474756150994399076829037077631004
1419104706752487096321682446755944831385669203205239183769600176398271132404285440311694032486384271360404552879664619585536398618427309988773888303825020815540224000180775887385246433280840863877053379379203039516145742133657684175152520973230081741211783826278400257833283374275840253926718474665601443258699022672332329305696011988727640913598577169
Technical details
Node id
n-f94383b851354fab · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-a9de0b27ed80361e

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:           312,            266,             49;
  1:          4752,           6552,           2793,             343;
  2:         69120,         133056,          91728,           26068,            2401;
  3:        974592,        2419200,        2328480,         1070160,          228095,           16807;
  4:      13436928,       40932864,       50803200,        32598720,        11236680,         1915998,         117649;
  5:     182145024,      658409472,     1002855168,       829785600,       399334320,       110119464,       15647317,         823543;
  6:    2436562944,    10200121344,    18435465216,     18719963136,     11616998400,      4472544384,     1027781664,      125178536,       5764801;
  7:   32248627200,   153503465472,   321303822336,    387144769536,    294839419392,    146374179840,    46961716032,     9250034976,     985780971,     40353607;
  8:  423101988864,  2257403904000,  5372621291520,   7497089187840,   6775033466880,   4127751871488,  1707698764800,   469617160320,   80937806040,   7667185330,   282475249;
  9: 5510645415936, 32578853142528, 86910050304000, 137897279815680, 144318966865920, 104335515389952, 52972815684096, 18784686412800, 4520065168080, 692467896120, 59037327041, 1977326743;
%F Formula
The row polynomials satisfy (7*n+19+7*x)*P(n,x) = (84*n+312+(49*n+266)*x+49*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 312 + 266*x + 49*x^2.
%t Mathematica
P[0, x_] := 312 + 266*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((84*n + 312 + (49*n + 266)*x + 49*x^2)*P[n - 1, x])/(7*n + 19 + 7*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 312 + 266*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((84*n + 312 + (49*n + 266)*x + 49*x^2)*v[i - 1])/(7*n + 19 + 7*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.