Real-rooted atlas

A352880 → reverse → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(3\,n + 10\,n^{2} + 9\,n^{3} + 2\,n^{4}\right)\,P_{n}(x) = \left(54\,x^{2} + 63\,n x^{2} - 36\,n^{2} x^{2} - 63\,n^{3} x^{2} - 18\,n^{4} x^{2}\right)\,P_{n-2}(x) + \left(12\,n + 38\,n^{2} + 48\,n^{2} x + 32\,n^{3} + 48\,n^{3} x + 8\,n^{4} + 12\,n^{4} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 8\)
\(P_{1}(x) = 30 + 36\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
088
1303666
2112288144544
3420168018905404530
4158486401555210368194438088
5600641580103950113400510306804322770
622880192192617760950400712800233280233282752640
787516864864340540268108047297290404157610103947873223596578

All 15 rows

Technical details
Node id
n-c0d268b8d0600f93 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-dae6101d606a19f7

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:       8;
  1:      30,       36;
  2:     112,      288,      144;
  3:     420,     1680,     1890,       540;
  4:    1584,     8640,    15552,     10368,      1944;
  5:    6006,    41580,   103950,    113400,     51030,      6804;
  6:   22880,   192192,   617760,    950400,    712800,    233280,     23328;
  7:   87516,   864864,  3405402,   6810804,   7297290,   4041576,   1010394,     78732;
  8:  335920,  3818880, 17821440,  44029440,  61916400,  49533120,  21228480,   4199040,   262440;
  9: 1293292, 16628040, 89791416, 264648384, 463134672, 490377888, 306486180, 105080976, 16888014, 866052;
%F Formula
The row polynomials satisfy (2*n^4+9*n^3+10*n^2+3*n)*P(n,x) = -(18*n^4+63*n^3+36*n^2-63*n-54)*x^2*P(n-2,x) + (8*n^4+32*n^3+38*n^2+12*n+(12*n^4+48*n^3+48*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 8, P(1,x) = 30 + 36*x.
%t Mathematica
P[0, x_] := 8;
P[1, x_] := 30 + 36*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n^4 + 63*n^3 + 36*n^2 - 63*n - 54)*x^2*P[n - 2, x] + (8*n^4 + 32*n^3 + 38*n^2 + 12*n + (12*n^4 + 48*n^3 + 48*n^2)*x)*P[n - 1, x])/(2*n^4 + 9*n^3 + 10*n^2 + 3*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 8; v[2] = 30 + 36*x; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n^4 + 63*n^3 + 36*n^2 - 63*n - 54)*x^2*v[i - 2] + (8*n^4 + 32*n^3 + 38*n^2 + 12*n + (12*n^4 + 48*n^3 + 48*n^2)*x)*v[i - 1])/(2*n^4 + 9*n^3 + 10*n^2 + 3*n)); 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.