Real-rooted atlas

A038322 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(11 + 8\,x + 8\,n\right)\,P_{n}(x) = \left(361 + 304\,x + 64\,x^{2} + 152\,n + 64\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
127835
2665432641161
31552315960518451237179
43498094967362553605529640961161297
5768893913992360993472034048005529603276835606547
61658986333690690723358166401589555204085760053084162621441076168025
735284410759290323448103339340166268577280222537728045760512049545216209715232155900587
87419135433722582022880029729035033622045725900810029723648028484829184488112128045298483216777216951892141473
915455042367315341777512264812952823680071349684080643968230662144144428020531234181795020850205818880407686348813421772827960524111619
103194608208012112364033893848021367110049056021678741964800014269936816128063491690594304192570694041603906490859520502058188800362387865601073741824815938930893609
1165602408958470728112552230506485440174913293120626768561438976047693232322560002511508879638528931211462049792242088872509440429713994547204909013401600318901321728858993459223677480190884635
121339637977329518562978312600131872134940250706431104174085597225379840150424454745354240915710060593152004018414207421644812770900050968576290506647011328045836159418368047126528655360278313880780868719476736683764449686329329
132722377350451378031393223496422699240327487225520685734446779286911562782724526225527859875840312882865870336819215872307716947968005970215393883586561660217006625914883356965698797568047669605795102724455599072870402412053633433654975581388819662397223906884851
14550894083355567472930490626325055433936780205157996711574401222618975277226741761309820033523757916161013874518240612188165840480162912953958425395692347116748800835830155143702118420660478304678051843759801582653276164853632590046822441585591346790402078076976496644398046511104563305166269637701113
Technical details
Node id
n-2cffd412c97091fc · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-107531ef935485d1

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:            27,             8;
  2:           665,           432,            64;
  3:         15523,         15960,          5184,           512;
  4:        349809,        496736,        255360,         55296,          4096;
  5:       7688939,      13992360,       9934720,       3404800,        552960,         32768;
  6:     165898633,     369069072,     335816640,     158955520,      40857600,       5308416,       262144;
  7:    3528441075,    9290323448,   10333934016,    6268577280,    2225377280,     457605120,     49545216,     2097152;
  8:   74191354337,  225820228800,  297290350336,  220457259008,  100297236480,   28484829184,   4881121280,   452984832,   16777216;
  9: 1545504236731, 5341777512264, 8129528236800, 7134968408064, 3968230662144, 1444280205312, 341817950208, 50205818880, 4076863488, 134217728;
%F Formula
The row polynomials satisfy (8*n+11+8*x)*P(n,x) = (152*n+361+(64*n+304)*x+64*x^2)*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[Together[((152*n + 361 + (64*n + 304)*x + 64*x^2)*P[n - 1, x])/(8*n + 11 + 8*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] = ((152*n + 361 + (64*n + 304)*x + 64*x^2)*v[i - 1])/(8*n + 11 + 8*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. [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.