Real-rooted atlas

A225470 → basis A013609 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 24\,x + 36\,x^{2} + 18\,n - 24\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-24\,x^{2} + 16\,x^{3} + 24\,n x^{2}\right)\,P_{n-2}'(x) + \left(-3 + 8\,x + 6\,n\right)\,P_{n-1}(x) + \left(-8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 3 + 8\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13811
2187236126
31627926481281730
419441080011340384040028324
52916017755221870097920180001152542484
652488034292164735584254016063000072576313611935552
71102248076212576114878736701809922116800033384962634248192297072896
826453952019172816643100046256209280153672734760014108774415410304884736207368259420096
9714256704053882921088923290164006752802816026178390000590413017680015040064143360279936051200253832197184
1021427701120016736281689603012343636896235412179200099637678800025251766848040076866368401448960024634368084480001239048547611337088
1170711413696005694394566528010693866677788888396574289408402370646040001120264820928020099457987842371687464961829101824088704000024532992294912313056358346880
1225456108930560021065511749068804106039759497728356250304518451217247736203984005191780815936001028541466592641378854553190412588100462087709644800030298245126900940869222412391588469098240
139927882482918400841919845358131201696150309074923521535368712404101127839899321173920025215293208897792542462983071609680643349558886483995847679744611493442560030473656012899028500481889771521605632527207544525674240
144169710642825728003615486410367498240750269522852583782470516242575460864003772560437519328000128483141522351616029647608440318668848040668413337600556911038825856046422440064000027597076835328114124308480031181230080505774080368640023994747671080872960
Technical details
Node id
n-32c301f0d6d2f563 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d3916796115e3626

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,           8;
  2:         18,          72,          36;
  3:        162,         792,         648,         128;
  4:       1944,       10800,       11340,        3840,         400;
  5:      29160,      177552,      218700,       97920,       18000,       1152;
  6:     524880,     3429216,     4735584,     2540160,      630000,      72576,      3136;
  7:   11022480,    76212576,   114878736,    70180992,    21168000,    3338496,    263424,     8192;
  8:  264539520,  1917281664,  3100046256,  2092801536,   727347600,  141087744,  15410304,   884736,   20736;
  9: 7142567040, 53882921088, 92329016400, 67528028160, 26178390000, 5904130176, 800150400, 64143360, 2799360, 51200;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(24*n-24)*x-36*x^2)*P(n-2,x) + ((24*n-24)*x^2+16*x^3)*d/dx P(n-2,x) + (6*n-3+8*x)*P(n-1,x) - 8*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 3 + 8*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 3 + 8*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (24*n - 24)*x - 36*x^2)*P[n - 2, x] + ((24*n - 24)*x^2 + 16*x^3)*D[P[n - 2, x], x] + (6*n - 3 + 8*x)*P[n - 1, x] - 8*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] = 3 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (24*n - 24)*x - 36*x^2)*v[i - 2] + ((24*n - 24)*x^2 + 16*x^3)*deriv(v[i - 2], x) + (6*n - 3 + 8*x)*v[i - 1] - 8*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

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