Real-rooted atlas

A133214 → basis A013609 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(4 + 36\,x + 64\,x^{2} + 2\,n + 34\,n x + 96\,n x^{2} + 8\,n^{2} x + 48\,n^{2} x^{2} + 8\,n^{3} x^{2}\right)\,P_{n}(x) = \left(20 + 252\,x + 896\,x^{2} + 832\,x^{3} + 14\,n + 270\,n x + 1232\,n x^{2} + 1312\,n x^{3} + 64\,n^{2} x + 496\,n^{2} x^{2} + 640\,n^{2} x^{3} + 56\,n^{3} x^{2} + 96\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-1 - 24\,x - 172\,x^{2} - 480\,x^{3} - 448\,x^{4} - 8\,n x - 92\,n x^{2} - 328\,n x^{3} - 352\,n x^{4} - 8\,n^{2} x^{2} - 48\,n^{2} x^{3} - 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 13 + 64\,x + 48\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0136448125
1634567202561495
2321307274886144128018305
31683200406624087040448006144225947
48989128064534960972800825600294912286722793997
54863980679240753449488640113344006838272180633613107234529495
6265729502988829809920846970881309952001128529925138022410485760589824426106625
71462563311092562116247047102341121351203840152196710410019143683586129925839257626214405249142955

All 15 rows

Technical details
Node id
n-8c5c5edccf3c0771 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-f9fc0a64807f736f

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 2)
  • 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:       13,         64,          48;
  1:       63,        456,         720,         256;
  2:      321,       3072,        7488,        6144,         1280;
  3:     1683,      20040,       66240,       87040,        44800,         6144;
  4:     8989,     128064,      534960,      972800,       825600,       294912,        28672;
  5:    48639,     806792,     4075344,     9488640,     11334400,      6838272,      1806336,       131072;
  6:   265729,    5029888,    29809920,    84697088,    130995200,    112852992,     51380224,     10485760,      589824;
  7:  1462563,   31109256,   211624704,   710234112,   1351203840,   1521967104,   1001914368,    358612992,    58392576,     2621440;
  8:  8097453,  191200320,  1468275120,  5685534720,  12853478400,  17935368192,  15570616320,   8147435520,  2362245120,   314572800,   11534336;
  9: 45046719, 1169171784, 10003744080, 43909536000, 115071052800, 191833399296, 207625076736, 144381050880, 61798809600, 14850457600, 1649410048, 50331648;
%F Formula
The row polynomials satisfy (2*n+4+(8*n^2+34*n+36)*x+(8*n^3+48*n^2+96*n+64)*x^2)*P(n,x) = (14*n+20+(64*n^2+270*n+252)*x+(56*n^3+496*n^2+1232*n+896)*x^2+(96*n^3+640*n^2+1312*n+832)*x^3)*P(n-1,x) - (1+(8*n+24)*x+(8*n^2+92*n+172)*x^2+(48*n^2+328*n+480)*x^3+(64*n^2+352*n+448)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 13 + 64*x + 48*x^2.
%t Mathematica
P[0, x_] := 13 + 64*x + 48*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((14*n + 20 + (64*n^2 + 270*n + 252)*x + (56*n^3 + 496*n^2 + 1232*n + 896)*x^2 + (96*n^3 + 640*n^2 + 1312*n + 832)*x^3)*P[n - 1, x] - (1 + (8*n + 24)*x + (8*n^2 + 92*n + 172)*x^2 + (48*n^2 + 328*n + 480)*x^3 + (64*n^2 + 352*n + 448)*x^4)*D[P[n - 1, x], x])/(2*n + 4 + (8*n^2 + 34*n + 36)*x + (8*n^3 + 48*n^2 + 96*n + 64)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 13 + 64*x + 48*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((14*n + 20 + (64*n^2 + 270*n + 252)*x + (56*n^3 + 496*n^2 + 1232*n + 896)*x^2 + (96*n^3 + 640*n^2 + 1312*n + 832)*x^3)*v[i - 1] - (1 + (8*n + 24)*x + (8*n^2 + 92*n + 172)*x^2 + (48*n^2 + 328*n + 480)*x^3 + (64*n^2 + 352*n + 448)*x^4)*deriv(v[i - 1], x))/(2*n + 4 + (8*n^2 + 34*n + 36)*x + (8*n^3 + 48*n^2 + 96*n + 64)*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