Real-rooted atlas

A116412 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(16 + 16\,n\right)\,P_{n}(x) = \left(64 + 20\,x^{2} + 26\,x^{3} + 64\,n + 64\,n x - 60\,n x^{2} - 39\,n x^{3} + 64\,n^{2} x + 40\,n^{2} x^{2} + 13\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(-64\,x + 96\,x^{2} + 204\,x^{3} + 52\,x^{4} - 64\,n x - 192\,n x^{2} - 132\,n x^{3} - 26\,n x^{4}\right)\,P_{n-1}'(x) + \left(64\,x^{2} + 176\,x^{3} + 92\,x^{4} + 13\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1448
216321361
36419216843467
4256102413927601423574
51024512094728080318846927353
6409624576576006732841424127681549209341
7163841146883256324851204094721961764956451161602152
86553652428817489923172352342016022461448792801881201689712261769
9262144235929690439681935360025463808213680641144972837847527019765580793843143
101048576104857604541644811206656017435443217871667212230784055251456157939202584864184318718210846
114194304461373442228224006230507521120075776135913881611363174406550963202555187846431387294170686087615496691637
12167772162013265921072693248335282176068431380489606266880951190732866972856323330677760114223897625674326434008384201060142067895689
1367108864872415232508349644817566793728401460428806405783552073423060992612292730883717127065616238160640496646425610082454721219196686640564321958728008
142684354563758096384237733150729000976384022776853299240739484467253113395609651284017152036867395174419648645120076479972608211015866883905191888434372048219322932464050574501
Technical details
Node id
n-4ea14f870cf61a3a · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-97d66d9ae239f251

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:      4,       4;
  2:     16,      32,      13;
  3:     64,     192,     168,       43;
  4:    256,    1024,    1392,      760,      142;
  5:   1024,    5120,    9472,     8080,     3188,      469;
  6:   4096,   24576,   57600,    67328,    41424,    12768,     1549;
  7:  16384,  114688,  325632,   485120,   409472,   196176,    49564,    5116;
  8:  65536,  524288, 1748992,  3172352,  3420160,  2246144,   879280,  188120,  16897;
  9: 262144, 2359296, 9043968, 19353600, 25463808, 21368064, 11449728, 3784752, 701976, 55807;
%F Formula
The row polynomials satisfy (16*n+16)*P(n,x) = (64*n+64+(64*n^2+64*n)*x+(40*n^2-60*n+20)*x^2+(13*n^2-39*n+26)*x^3)*P(n-1,x) - ((64*n+64)*x+(192*n-96)*x^2+(132*n-204)*x^3+(26*n-52)*x^4)*d/dx P(n-1,x) + (64*x^2+176*x^3+92*x^4+13*x^5)*d^2/dx^2 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[((64*n + 64 + (64*n^2 + 64*n)*x + (40*n^2 - 60*n + 20)*x^2 + (13*n^2 - 39*n + 26)*x^3)*P[n - 1, x] - ((64*n + 64)*x + (192*n - 96)*x^2 + (132*n - 204)*x^3 + (26*n - 52)*x^4)*D[P[n - 1, x], x] + (64*x^2 + 176*x^3 + 92*x^4 + 13*x^5)*D[P[n - 1, x], {x, 2}])/(16*n + 16)]];
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] = ((64*n + 64 + (64*n^2 + 64*n)*x + (40*n^2 - 60*n + 20)*x^2 + (13*n^2 - 39*n + 26)*x^3)*v[i - 1] - ((64*n + 64)*x + (192*n - 96)*x^2 + (132*n - 204)*x^3 + (26*n - 52)*x^4)*deriv(v[i - 1], x) + (64*x^2 + 176*x^3 + 92*x^4 + 13*x^5)*deriv(deriv(v[i - 1], x), x))/(16*n + 16)); 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. [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