Real-rooted atlas

A114687 → basis A013611 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 64\,x + 4\,n + 96\,n x + 128\,n x^{2} + 32\,n^{2} x + 192\,n^{2} x^{2} + 64\,n^{3} x^{2}\right)\,P_{n}(x) = \left(8 + 64\,x + 28\,n + 448\,n x + 1664\,n x^{2} + 1024\,n x^{3} + 256\,n^{2} x + 2112\,n^{2} x^{2} + 3584\,n^{2} x^{3} + 448\,n^{3} x^{2} + 1536\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 28\,x - 256\,x^{2} - 896\,x^{3} - 1024\,x^{4} - 16\,n x - 272\,n x^{2} - 1472\,n x^{3} - 2560\,n x^{4} - 32\,n^{2} x^{2} - 384\,n^{2} x^{3} - 1024\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
131619
211112192315
345672230420485069
4197387219968368642048081381
5903220001516804300805120001966081313271
642791243681074240412672076800006488064183500821332679
720793702016728985635409920903168001211105287707033616777216348697465

All 15 rows

Technical details
Node id
n-ee5803cd08e67ef2 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-376230ec75b31fe8

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,       16;
  2:     11,      112,       192;
  3:     45,      672,      2304,       2048;
  4:    197,     3872,     19968,      36864,      20480;
  5:    903,    22000,    151680,     430080,     512000,      196608;
  6:   4279,   124368,   1074240,    4126720,    7680000,     6488064,     1835008;
  7:  20793,   702016,   7289856,   35409920,   90316800,   121110528,    77070336,    16777216;
  8: 103049,  3962944,  48082944,  282705920,  919224320,  1706557440,  1746927616,   872415232,  150994944;
  9: 518859, 22388688, 310839552, 2146615296, 8500101120, 20263993344, 28978446336, 23555211264, 9512681472, 1342177280;
%F Formula
The row polynomials satisfy (4*n+8+(32*n^2+96*n+64)*x+(64*n^3+192*n^2+128*n)*x^2)*P(n,x) = (28*n+8+(256*n^2+448*n+64)*x+(448*n^3+2112*n^2+1664*n)*x^2+(1536*n^3+3584*n^2+1024*n)*x^3)*P(n-1,x) - (1+(16*n+28)*x+(32*n^2+272*n+256)*x^2+(384*n^2+1472*n+896)*x^3+(1024*n^2+2560*n+1024)*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[((28*n + 8 + (256*n^2 + 448*n + 64)*x + (448*n^3 + 2112*n^2 + 1664*n)*x^2 + (1536*n^3 + 3584*n^2 + 1024*n)*x^3)*P[n - 1, x] - (1 + (16*n + 28)*x + (32*n^2 + 272*n + 256)*x^2 + (384*n^2 + 1472*n + 896)*x^3 + (1024*n^2 + 2560*n + 1024)*x^4)*D[P[n - 1, x], x])/(4*n + 8 + (32*n^2 + 96*n + 64)*x + (64*n^3 + 192*n^2 + 128*n)*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] = ((28*n + 8 + (256*n^2 + 448*n + 64)*x + (448*n^3 + 2112*n^2 + 1664*n)*x^2 + (1536*n^3 + 3584*n^2 + 1024*n)*x^3)*v[i - 1] - (1 + (16*n + 28)*x + (32*n^2 + 272*n + 256)*x^2 + (384*n^2 + 1472*n + 896)*x^3 + (1024*n^2 + 2560*n + 1024)*x^4)*deriv(v[i - 1], x))/(4*n + 8 + (32*n^2 + 96*n + 64)*x + (64*n^3 + 192*n^2 + 128*n)*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

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