Real-rooted atlas

A038220 → basis A007318 → D+1

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(3 + 2\,x + 2\,n\right)\,P_{n}(x) = \left(25 + 20\,x + 4\,x^{2} + 10\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1729
24528477
3275270848637
4162522001080224165145
59375162501100036005603240817
653125112500975004400010800134464319333
72968757437507875004550001540003024031361282470629
81640625475000059500004200000182000049280080640716825618941489
98984375295312504275000035700000189000006552000147840020736016128512144120025
1048828125179687500295312500285000000178500000756000002184000042240005184003584010241089547389
112636718751074218750197656250021656250001567500000785400000277200000686400001161600012672007884820488191782221
121416015625632812500012890625000158125000001299375000075240000003141600000950400000205920000309760003041280172032409661297129033
137568359375368164062508226562500011171875000010278125000067567500000326040000001166880000030888000005948800008053760071884803727368192456762477633
1440283203125211914062500515429687500767812500000782031250000575575000000315315000000130416000000408408000009609600000166566400020500480016773120802816163843391115364245
Technical details
Node id
n-5cfb4e705168cb18 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b29ddf49d7bb6a39

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:       7,        2;
  2:      45,       28,        4;
  3:     275,      270,       84,        8;
  4:    1625,     2200,     1080,      224,       16;
  5:    9375,    16250,    11000,     3600,      560,      32;
  6:   53125,   112500,    97500,    44000,    10800,    1344,      64;
  7:  296875,   743750,   787500,   455000,   154000,   30240,    3136,    128;
  8: 1640625,  4750000,  5950000,  4200000,  1820000,  492800,   80640,   7168,   256;
  9: 8984375, 29531250, 42750000, 35700000, 18900000, 6552000, 1478400, 207360, 16128, 512;
%F Formula
The row polynomials satisfy (2*n+3+2*x)*P(n,x) = (10*n+25+(4*n+20)*x+4*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[((10*n + 25 + (4*n + 20)*x + 4*x^2)*P[n - 1, x])/(2*n + 3 + 2*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] = ((10*n + 25 + (4*n + 20)*x + 4*x^2)*v[i - 1])/(2*n + 3 + 2*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

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.