Real-rooted atlas

A166341 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-8\,n x + 16\,n x^{2} - 4\,n^{2} x + 25\,n^{2} x^{2} + 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-4\,x + 17\,x^{2} - 23\,x^{3} - 2\,n x + 16\,n x^{2} - 29\,n x^{3} + 4\,n^{2} x^{2} - 2\,n^{2} x^{3} + 4\,n^{3} x^{3}\right)\,P_{n-2}'(x) + \left(-2\,x^{2} + 9\,x^{3} - 7\,x^{4} - n x^{2} + 12\,n x^{3} - 11\,n x^{4} + 4\,n^{2} x^{3} - 4\,n^{2} x^{4}\right)\,P_{n-2}''(x) + \left(1 + 4\,x + n + 8\,n x + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 8\,x^{2} + 2\,n x - 12\,n x^{2} - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 40\,x + 9\,x^{2}\)
\(P_{2}(x) = 1 + 92\,x + 207\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
1140950
219220716316
312001242800252268
414205526982426253618432
51864213037859259175777649167760
6117567557350372878707530229221511641692000
7135442543762821024787075063473041384936567048118738720
817124826920144496646580085094753224442520965880320144261100226114560
91142882625021695592964870474501137038688954613002278237184236251893572001212953117440
10128620819654332023752033047284501173167042415968106966846010483216212024459107270086575514441506214400
11157288252979741426666400210269823751084604942882235534975241928186565126812742289589166650003401172062062368169624697920000
121114628774406986201381536127426574375923272657308274588875380435864669437442077363478943509706297500468979478661239051168484303319610024545484800
131229312235674963264575943687437191170257374578614848305313113286515738347109990450470126890219204849405968003599600526401238118349312442545208311236288225170856996710400
141458684714248217111277087728421467547952556027982502764314095388224877817966448907456103523878689849964101099637730018831627452317924276530762064117536423912715554738948258009752563082450020249600
1519174322158225236462885195424233336932421004089974873027763037531122268300106462019680743681863429816417298216634690575116200750084215907070016359899492111041577357663165965670343643944539556309005871564828958685810287104000
Technical details
Node id
n-49f893754ca98ec1 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-17c9767dee58f463

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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,    40,       9;
   2: 1,    92,     207,        16;
   3: 1,   200,    1242,       800,         25;
   4: 1,   420,    5526,      9824,       2625,          36;
   5: 1,   864,   21303,     78592,      59175,        7776,          49;
   6: 1,  1756,   75573,    503728,     787075,      302292,       21511,         64;
   7: 1,  3544,  254376,   2821024,    7870750,     6347304,     1384936,      56704,         81;
   8: 1,  7124,  826920,  14449664,   65800850,    94753224,    44252096,    5880320,     144261,      100;
   9: 1, 14288, 2625021,  69559296,  487047450,  1137038688,   954613002,  278237184,   23625189,   357200,    121;
  10: 1, 28620, 8196543, 320237520, 3304728450, 11731670424, 15968106966, 8460104832, 1621202445, 91072700, 865755, 144;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -((4*n^2+8*n)*x-(9*n^3+25*n^2+16*n)*x^2)*P(n-2,x) - ((2*n+4)*x-(4*n^2+16*n+17)*x^2-(4*n^3-2*n^2-29*n-23)*x^3)*d/dx P(n-2,x) - ((n+2)*x^2-(4*n^2+12*n+9)*x^3+(4*n^2+11*n+7)*x^4)*d^2/dx^2 P(n-2,x) + (n+1+(4*n^2+8*n+4)*x)*P(n-1,x) + ((2*n+3)*x-(4*n^2+12*n+8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 40*x + 9*x^2, P(2,x) = 1 + 92*x + 207*x^2 + 16*x^3.
%t Mathematica
P[1, x_] := 1 + 40*x + 9*x^2;
P[2, x_] := 1 + 92*x + 207*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*P[n - 2, x] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*D[P[n - 2, x], x] - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*D[P[n - 2, x], {x, 2}] + (n + 1 + (4*n^2 + 8*n + 4)*x)*P[n - 1, x] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*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 + 40*x + 9*x^2; v[2] = 1 + 92*x + 207*x^2 + 16*x^3; for(n = 3, N, my(i = n); v[i] = (-((4*n^2 + 8*n)*x - (9*n^3 + 25*n^2 + 16*n)*x^2)*v[i - 2] - ((2*n + 4)*x - (4*n^2 + 16*n + 17)*x^2 - (4*n^3 - 2*n^2 - 29*n - 23)*x^3)*deriv(v[i - 2], x) - ((n + 2)*x^2 - (4*n^2 + 12*n + 9)*x^3 + (4*n^2 + 11*n + 7)*x^4)*deriv(deriv(v[i - 2], x), x) + (n + 1 + (4*n^2 + 8*n + 4)*x)*v[i - 1] + ((2*n + 3)*x - (4*n^2 + 12*n + 8)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.