Real-rooted atlas

A106246 → basis A007318 → basis A013611

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 48\,x + 96\,x^{2} + 64\,x^{3}\right)\,P_{n-3}(x) + \left(-12 - 48\,x - 48\,x^{2}\right)\,P_{n-2}(x) + \left(6 + 12\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + 4\,x\)
\(P_{2}(x) = 13 + 32\,x + 16\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
213321661
33815619264450
4104608124810242563240
52722080608083205120102422896
66886528249604864049920245764096159408
716961926491392232960340480279552114688163841096416
8409654272308224974848186368021790721490944524288655367464960
997281474569768963698688877363213418496130744327667712235929626214450388480
10227843891202949120130252803698688070189056894566407471104038338560104857601048576337602816
115273610024968560640432537601432780803254845445147197445622988804109107201874329604613734441943042247326208
12120832253132824059904136970240519045120137546956826038763523529506816337379328021915238408996782082013265921677721614874679296
1327443262832646581452841703833617806131205398069248119207362561934308147222941794304194930278401139592396842530242568724152326710886497955205120
14618496153681921759313921228537856583853670419942866944503819796489536589004813540157030414274894233610916095590458015612928198474465283758096384268435456642150789120
Technical details
Node id
n-30a87cf8ff1909c0 · revision 0 · JSON
Definition source
derived:basis_map_A013611
Fingerprint
f6-3e4f17e08687f0a2

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:    1;
  1:    4,      4;
  2:   13,     32,     16;
  3:   38,    156,    192,      64;
  4:  104,    608,   1248,    1024,     256;
  5:  272,   2080,   6080,    8320,    5120,     1024;
  6:  688,   6528,  24960,   48640,   49920,    24576,     4096;
  7: 1696,  19264,  91392,  232960,  340480,   279552,   114688,   16384;
  8: 4096,  54272, 308224,  974848, 1863680,  2179072,  1490944,  524288,   65536;
  9: 9728, 147456, 976896, 3698688, 8773632, 13418496, 13074432, 7667712, 2359296, 262144;
%F Formula
The row polynomials satisfy P(n,x) = (8+48*x+96*x^2+64*x^3)*P(n-3,x) - (12+48*x+48*x^2)*P(n-2,x) + (6+12*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + 4*x, P(2,x) = 13 + 32*x + 16*x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + 4*x;
P[2, x_] := 13 + 32*x + 16*x^2;
P[n_, x_] := P[n, x] = Expand[(8 + 48*x + 96*x^2 + 64*x^3)*P[n - 3, x] - (12 + 48*x + 48*x^2)*P[n - 2, x] + (6 + 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; v[2] = 4 + 4*x; v[3] = 13 + 32*x + 16*x^2; for(n = 3, N, my(i = n + 1); v[i] = (8 + 48*x + 96*x^2 + 64*x^3)*v[i - 3] - (12 + 48*x + 48*x^2)*v[i - 2] + (6 + 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.