Real-rooted atlas

A134991 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-2\,n x^{2} - 4\,n^{2} x - 7\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-x^{2} - 3\,x^{3} - 2\,n x - 8\,n x^{2} - 11\,n x^{3} + 4\,n^{2} x^{2} + 10\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-x^{3} - x^{4} - n x^{2} - 4\,n x^{3} - 3\,n x^{4} + 4\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(-1 - 4\,x + n + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + 2\,n x + 4\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
211213
3140135176
4110094516802726
512244410201602362548420
6147617262151200433125374220976284
719846142591168047643759729720662161522089800
812004206415484792040915875147567420231756525129729600555025760
91404866884423795200303052750171026856046351305005881075200279121342515345208528
101814021158281106385282035783750168075507606969176214014996741760015909916522565472907500463187349472
11116328658557949521472012772009250147749762160876750384510283820689152050381402321254583103525000166366657957515160891200768
1213270820271069215647432076240414250119915735940097571218644904446337896000011718537403072517644948571250014030254821082545537716624400534973199954688
131654726191710292024965764384834203759171095693760993113947625806105869497728002236421610197775494730748512000064539172176979504553771662440000133608925512562520247024418710016
1411310041881082983868125872024512373180756702348054402094483736833486076086652830988803712882487451637511297640452348250021466289876256225024666263171550000015632244284969812541837777148667500818213996113220608
1512620725693357971607445004161340453645862547273651802468085296160638939258801893480796416055564291484585383522298773748501250005778828123223530375960208292742098400098483138995309818755662045840786335000139281375456660937535166639688464859136
Technical details
Node id
n-efeec4489ace5743 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-9bbe3a476ec7224d

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,   12;
   3: 1,   40,     135;
   4: 1,  100,     945,      1680;
   5: 1,  224,    4410,     20160,      23625;
   6: 1,  476,   17262,    151200,     433125,      374220;
   7: 1,  984,   61425,    911680,    4764375,     9729720,     6621615;
   8: 1, 2004,  206415,   4847920,   40915875,   147567420,   231756525,    129729600;
   9: 1, 4048,  668844,  23795200,  303052750,  1710268560,  4635130500,   5881075200,   2791213425;
  10: 1, 8140, 2115828, 110638528, 2035783750, 16807550760, 69691762140, 149967417600, 159099165225, 65472907500;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^2*x-(9*n^3-7*n^2-2*n)*x^2)*P(n-2,x) - (2*n*x-(4*n^2-8*n-1)*x^2-(4*n^3+10*n^2-11*n-3)*x^3)*d/dx P(n-2,x) - (n*x^2-(4*n^2-4*n-1)*x^3-(4*n^2-3*n-1)*x^4)*d^2/dx^2 P(n-2,x) + (n-1+(4*n^2-4)*x)*P(n-1,x) + ((2*n-1)*x-(4*n^2-4*n)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 12*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2*x - (9*n^3 - 7*n^2 - 2*n)*x^2)*P[n - 2, x] - (2*n*x - (4*n^2 - 8*n - 1)*x^2 - (4*n^3 + 10*n^2 - 11*n - 3)*x^3)*D[P[n - 2, x], x] - (n*x^2 - (4*n^2 - 4*n - 1)*x^3 - (4*n^2 - 3*n - 1)*x^4)*D[P[n - 2, x], {x, 2}] + (n - 1 + (4*n^2 - 4)*x)*P[n - 1, x] + ((2*n - 1)*x - (4*n^2 - 4*n)*x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 12*x; for(n = 3, N, my(i = n); v[i] = (-(4*n^2*x - (9*n^3 - 7*n^2 - 2*n)*x^2)*v[i - 2] - (2*n*x - (4*n^2 - 8*n - 1)*x^2 - (4*n^3 + 10*n^2 - 11*n - 3)*x^3)*deriv(v[i - 2], x) - (n*x^2 - (4*n^2 - 4*n - 1)*x^3 - (4*n^2 - 3*n - 1)*x^4)*deriv(deriv(v[i - 2], x), x) + (n - 1 + (4*n^2 - 4)*x)*v[i - 1] + ((2*n - 1)*x - (4*n^2 - 4*n)*x^2)*deriv(v[i - 1], x))/(n - 1)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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