Real-rooted atlas

A136124 → basis A008277 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 - x + 2\,n x + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 2\,x + x^{2} + x^{3} + n + 3\,n x + 3\,n x^{2} + 2\,n x^{3} + 2\,n^{2} x + 2\,n^{2} x^{2} + n^{2} x^{3} + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(x + x^{2} + x^{3} + 2\,n x^{2} + 2\,n x^{3} + n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1347
21232953
36024013516451
436019201620384254309
52520168001890067208753645851
62016016128022680010752021000172849538537
7181440169344028576801693440441000544323087646924583
818144001935360038102400270950408820000145152012348051208196765641
91995840023950080053887680044706816017463600035925120407484025344080191001460301679
102395008003193344000808315200076640256003492720000862202880122245200101376004811401200012123667821341
11311351040045664819200128432304000136994457600713512800002054916864034962127203624192002293434085800017303144409987981547
12435891456006974263296002157662707200255722987520014983768800004931800473609789395616012177285120963242280480480001453452241921697558548994333
136538371840001133317785600038249475264000498659825664003246483240000012021263654400272704592160039576176640037566448920234234000094474380235872032955196147751382267971
141046139494400019527937228800071399020492800010154527358976007272122457600002991958953984007635728580480012664376524800140248075968010493683200052905652801761177603690960439042253052126399555409
Technical details
Node id
n-060005e4e8c6f4b8 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-8360a1098ab5c82c

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:        3,         4;
  2:       12,        32,         9;
  3:       60,       240,       135,        16;
  4:      360,      1920,      1620,       384,        25;
  5:     2520,     16800,     18900,      6720,       875,       36;
  6:    20160,    161280,    226800,    107520,     21000,     1728,      49;
  7:   181440,   1693440,   2857680,   1693440,    441000,    54432,    3087,     64;
  8:  1814400,  19353600,  38102400,  27095040,   8820000,  1451520,  123480,   5120,   81;
  9: 19958400, 239500800, 538876800, 447068160, 174636000, 35925120, 4074840, 253440, 8019, 100;
%F Formula
The row polynomials satisfy (1+(2*n-1)*x+n^2*x^2)*P(n,x) = (n+2+(2*n^2+3*n+2)*x+(n^3+2*n^2+3*n+1)*x^2+(n^2+2*n+1)*x^3)*P(n-1,x) + (x+(2*n+1)*x^2+(n^2+2*n+1)*x^3)*d/dx 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[((n + 2 + (2*n^2 + 3*n + 2)*x + (n^3 + 2*n^2 + 3*n + 1)*x^2 + (n^2 + 2*n + 1)*x^3)*P[n - 1, x] + (x + (2*n + 1)*x^2 + (n^2 + 2*n + 1)*x^3)*D[P[n - 1, x], x])/(1 + (2*n - 1)*x + n^2*x^2)]];
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] = ((n + 2 + (2*n^2 + 3*n + 2)*x + (n^3 + 2*n^2 + 3*n + 1)*x^2 + (n^2 + 2*n + 1)*x^3)*v[i - 1] + (x + (2*n + 1)*x^2 + (n^2 + 2*n + 1)*x^3)*deriv(v[i - 1], x))/(1 + (2*n - 1)*x + n^2*x^2)); 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. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [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
  4. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.