Real-rooted atlas

A147716 → basis A013611 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(19 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1191231
2361456144961
36859129968208172829791
413032132923231190413132820736923521
52476099781926098769606238080196992024883228629151
647045881178279128281493360237047040112285440283668482985984887503681
7893871739395185400474877233767881814080497798784018863953923971358723583180827512614111
8169835630418581168694418968899219223960714803218916353792095577366528301823262725446434816429981696852891037441
932268769777918342248084284633831094976682880371891264693929968644085932419072172039259750446567017676873526870016515978035226439622160671
106131066257801387225237334801100534885056801853532437990402048641115673601552654319247368171864838144029492444528640698505265152098035826688061917364224819628286980801
111164902588982198093007460297322555686566409680484235349424992061166570453683205408412545378304341583950234419215409802266214404866253347225601024474388889601294072912281674300837068825408476896404831
12221331491906616116774597281343536582696537141407041226729551876646401743247257929971201761597229066076161298019010890792967026869833393766427737644079185920778600535556096014752431200010241694059085168648916100448256787662783788549761
1342052983462257059345277127374321116130841858794479580830300219931353166084784245252318920960543893144474151014445801527955717980162892728081413767168137023961751178444848078583070588928012146168354675097620921629701832704220227681071923210699320537907224417546297445042591
1479900668578288412170649012216591859122900327869944297374473271440924908565248127260923711683297536160750640477915744256152290080452762284032109923667093723152384607472897096891105282557780619355330969680772019558589399041855051166895833088292902815825657856284601926308331521283918464548864756943935220796320321
Technical details
Node id
n-d48d356b850c3f66 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b68b92f4c09b2b20

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:            1;
  1:           19,            12;
  2:          361,           456,           144;
  3:         6859,         12996,          8208,          1728;
  4:       130321,        329232,        311904,        131328,         20736;
  5:      2476099,       7819260,       9876960,       6238080,       1969920,        248832;
  6:     47045881,     178279128,     281493360,     237047040,     112285440,      28366848,       2985984;
  7:    893871739,    3951854004,    7487723376,    7881814080,    4977987840,    1886395392,     397135872,     35831808;
  8:  16983563041,   85811686944,  189688992192,  239607148032,  189163537920,   95577366528,   30182326272,   5446434816,   429981696;
  9: 322687697779, 1834224808428, 4633831094976, 6828803718912, 6469392996864, 4085932419072, 1720392597504, 465670176768, 73526870016, 5159780352;
%F Formula
The row polynomials satisfy P(n,x) = (19+12*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(19 + 12*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (19 + 12*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. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.