Real-rooted atlas

A001263 → basis A013611 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(2 - 4\,x + 5\,n + 4\,n^{2} + 19\,n^{2} x + 8\,n^{2} x^{2} + n^{3} + 21\,n^{3} x + 20\,n^{3} x^{2} + 6\,n^{4} x + 16\,n^{4} x^{2} + 4\,n^{5} x^{2}\right)\,P_{n}(x) = \left(2 - 4\,x + 8\,n + 42\,n x + 32\,n x^{3} + 10\,n^{2} + 118\,n^{2} x + 232\,n^{2} x^{2} + 144\,n^{2} x^{3} + 4\,n^{3} + 96\,n^{3} x + 360\,n^{3} x^{2} + 320\,n^{3} x^{3} + 24\,n^{4} x + 160\,n^{4} x^{2} + 224\,n^{4} x^{3} + 16\,n^{5} x^{2} + 48\,n^{5} x^{3}\right)\,P_{n-1}(x) + \left(-2\,x - 14\,x^{2} - 32\,x^{3} - 32\,x^{4} - 4\,n x - 47\,n x^{2} - 160\,n x^{3} - 144\,n x^{4} - 2\,n^{2} x - 44\,n^{2} x^{2} - 208\,n^{2} x^{3} - 256\,n^{2} x^{4} - 12\,n^{3} x^{2} - 88\,n^{3} x^{3} - 160\,n^{3} x^{4} - 8\,n^{4} x^{3} - 32\,n^{4} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
121618
2580144229
314336129610242670
4421344806414336640030186
513252804320012288012800036864336356
64292059221384084480014400009953282007043715693
71430800801009008512512012320000141926407024640104857640801494
8486231116846126082870067289689600147603456123633664461373445308416446001790
91679612093122062195215207628858705920012680478721534181376981467136286654464262144004857548796
10587864702880906984007739596803564288000958080614415341813760143130624007245987840170393600012687769652746191586
112080121830505639395664038201856002046528000066012807168132411654144164710318080122624409600504627200009769582592603979776571293406668
12742900713184001694525184184096563201125590400004243680460801026865889280160945510809616081315430409804185600003349571174405435817984028353495046174125076084
13267444027814176072316857608702746624059831383040025847157473287342091108352139486109368321765371071692814529803059200740255229542421356724879362948763484161315333734466598039836360
149694845108582264030665129040404974402560309355446400015077508526080492531945185281102812052193281699169656504321779900874752001230674319114245325833266790413158857048064156524714393660397977600717159517651581
Technical details
Node id
n-b991dcbedf3c1d83 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-777088b2ee260e77

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:     2,      16;
  2:     5,      80,      144;
  3:    14,     336,     1296,      1024;
  4:    42,    1344,     8064,     14336,      6400;
  5:   132,    5280,    43200,    122880,    128000,      36864;
  6:   429,   20592,   213840,    844800,   1440000,     995328,     200704;
  7:  1430,   80080,  1009008,   5125120,  12320000,   14192640,    7024640,   1048576;
  8:  4862,  311168,  4612608,  28700672,  89689600,  147603456,  123633664,  46137344,   5308416;
  9: 16796, 1209312, 20621952, 152076288, 587059200, 1268047872, 1534181376, 981467136, 286654464, 26214400;
%F Formula
The row polynomials satisfy (n^3+4*n^2+5*n+2+(6*n^4+21*n^3+19*n^2-4)*x+(4*n^5+16*n^4+20*n^3+8*n^2)*x^2)*P(n,x) = (4*n^3+10*n^2+8*n+2+(24*n^4+96*n^3+118*n^2+42*n-4)*x+(16*n^5+160*n^4+360*n^3+232*n^2)*x^2+(48*n^5+224*n^4+320*n^3+144*n^2+32*n)*x^3)*P(n-1,x) - ((2*n^2+4*n+2)*x+(12*n^3+44*n^2+47*n+14)*x^2+(8*n^4+88*n^3+208*n^2+160*n+32)*x^3+(32*n^4+160*n^3+256*n^2+144*n+32)*x^4)*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[((4*n^3 + 10*n^2 + 8*n + 2 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (16*n^5 + 160*n^4 + 360*n^3 + 232*n^2)*x^2 + (48*n^5 + 224*n^4 + 320*n^3 + 144*n^2 + 32*n)*x^3)*P[n - 1, x] - ((2*n^2 + 4*n + 2)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (8*n^4 + 88*n^3 + 208*n^2 + 160*n + 32)*x^3 + (32*n^4 + 160*n^3 + 256*n^2 + 144*n + 32)*x^4)*D[P[n - 1, x], x])/(n^3 + 4*n^2 + 5*n + 2 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (4*n^5 + 16*n^4 + 20*n^3 + 8*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] = ((4*n^3 + 10*n^2 + 8*n + 2 + (24*n^4 + 96*n^3 + 118*n^2 + 42*n - 4)*x + (16*n^5 + 160*n^4 + 360*n^3 + 232*n^2)*x^2 + (48*n^5 + 224*n^4 + 320*n^3 + 144*n^2 + 32*n)*x^3)*v[i - 1] - ((2*n^2 + 4*n + 2)*x + (12*n^3 + 44*n^2 + 47*n + 14)*x^2 + (8*n^4 + 88*n^3 + 208*n^2 + 160*n + 32)*x^3 + (32*n^4 + 160*n^3 + 256*n^2 + 144*n + 32)*x^4)*deriv(v[i - 1], x))/(n^3 + 4*n^2 + 5*n + 2 + (6*n^4 + 21*n^3 + 19*n^2 - 4)*x + (4*n^5 + 16*n^4 + 20*n^3 + 8*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023