Real-rooted atlas

A048993 → (θ+1)² → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 24\,x - 6\,x^{2} + 4\,n - 20\,n x + 5\,n x^{2} + 4\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 24\,x^{2} - 6\,x^{3} - 8\,n x^{2} + 2\,n x^{3}\right)\,P_{n-2}'(x) + \left(4\,x^{3} - x^{4}\right)\,P_{n-2}''(x) + \left(8 - 3\,x - 4\,n + 2\,n x\right)\,P_{n-1}(x) + \left(4\,x - 2\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
011
144
29413
31627447
42596634188
5362504001354825
6495401625144027943937
7641029504087504816567420270
88117921303437800425251545611434111835
91002916295681296542502361942504840022954657423
1012145006075037632011185231530900852625149280459944097622
11144665511550096228040951688794863888228036437504560169207426965867
121699504206305222750012881187401533446485894849658400152875251384416184234186685725
13196131823500644758325359502001538685723657871364567442882703060366331325041860003685541355314108
142251783656902395135049108299251351300016938979203157073920310838227714427012602597936251262352073719410289242825
Technical details
Node id
n-900aa85ccaa9b50d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-aa231a1ef2aefcc4

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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;
  2:   9,    4;
  3:  16,   27,     4;
  4:  25,   96,    63,      4;
  5:  36,  250,   400,    135,      4;
  6:  49,  540,  1625,   1440,    279,      4;
  7:  64, 1029,  5040,   8750,   4816,    567,     4;
  8:  81, 1792, 13034,  37800,  42525,  15456,  1143,    4;
  9: 100, 2916, 29568, 129654, 250236, 194250, 48400, 2295, 4;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+1+(4*n^2-20*n+24)*x-(n^2-5*n+6)*x^2)*P(n-2,x) - (4*x+(8*n-24)*x^2-(2*n-6)*x^3)*d/dx P(n-2,x) + (4*x^3-x^4)*d^2/dx^2 P(n-2,x) - (4*n-8-(2*n-3)*x)*P(n-1,x) + (4*x-2*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[(4*n + 1 + (4*n^2 - 20*n + 24)*x - (n^2 - 5*n + 6)*x^2)*P[n - 2, x] - (4*x + (8*n - 24)*x^2 - (2*n - 6)*x^3)*D[P[n - 2, x], x] + (4*x^3 - x^4)*D[P[n - 2, x], {x, 2}] - (4*n - 8 - (2*n - 3)*x)*P[n - 1, x] + (4*x - 2*x^2)*D[P[n - 1, x], 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; for(n = 2, N, my(i = n + 1); v[i] = (4*n + 1 + (4*n^2 - 20*n + 24)*x - (n^2 - 5*n + 6)*x^2)*v[i - 2] - (4*x + (8*n - 24)*x^2 - (2*n - 6)*x^3)*deriv(v[i - 2], x) + (4*x^3 - x^4)*deriv(deriv(v[i - 2], x), x) - (4*n - 8 - (2*n - 3)*x)*v[i - 1] + (4*x - 2*x^2)*deriv(v[i - 1], 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  8. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.