Real-rooted atlas

A121574 → basis A013611 → reverse

nonnegative coefficients

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

Recurrence

\(\left(16\,n\right)\,P_{n}(x) = \left(72\,x^{2} + 26\,x^{3} + 64\,n - 16\,n x - 112\,n x^{2} - 39\,n x^{3} + 64\,n^{2} x + 40\,n^{2} x^{2} + 13\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(256\,x^{2} + 256\,x^{3} + 52\,x^{4} - 64\,n x - 192\,n x^{2} - 132\,n x^{3} - 26\,n x^{4}\right)\,P_{n-1}'(x) + \left(64\,x^{2} + 176\,x^{3} + 92\,x^{4} + 13\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1437
216281054
36417614033413
425696012166201093161
51024486485126864256836024192
6409623552527365881634560102001189185149
7163841105923020804323843506561616163936439271417003
86553650790416384002870272298777618954887176641487561297010844766
926214422937608536064177152002259353618380288955424030670885532204283782998377
101048576102236164312268810353049615663923215612313610392755245697216127268322031644141481635212469
11419430445088768212598784579928064101654528012024995849801943045511687682098215685158704073854244672804861479168
1216777216197132288102760448031402229766263209984858972160083094077445717091328277950899293241740820515622426622960154332137206416521
13671088648556380164886364160165391892483700581990457794625536648333393925291986534431454179328134586516484034046848803089248952967085097243284752311487
142684354563690987520229176770568512339968021122934374437038902476847333933056044800683212831575408640016503227187263021320960170675398403102102640339075340168350502179298263014
Technical details
Node id
n-9b1b9cd2f87671d4 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-e80d933944530b2f

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,       3;
  2:     16,      28,      10;
  3:     64,     176,     140,       33;
  4:    256,     960,    1216,      620,      109;
  5:   1024,    4864,    8512,     6864,     2568,      360;
  6:   4096,   23552,   52736,    58816,    34560,    10200,    1189;
  7:  16384,  110592,  302080,   432384,   350656,   161616,   39364,    3927;
  8:  65536,  507904, 1638400,  2870272,  2987776,  1895488,  717664,  148756,  12970;
  9: 262144, 2293760, 8536064, 17715200, 22593536, 18380288, 9554240, 3067088, 553220, 42837;
%F Formula
The row polynomials satisfy 16*n*P(n,x) = (64*n+(64*n^2-16*n)*x+(40*n^2-112*n+72)*x^2+(13*n^2-39*n+26)*x^3)*P(n-1,x) - (64*n*x+(192*n-256)*x^2+(132*n-256)*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*n^2 - 16*n)*x + (40*n^2 - 112*n + 72)*x^2 + (13*n^2 - 39*n + 26)*x^3)*P[n - 1, x] - (64*n*x + (192*n - 256)*x^2 + (132*n - 256)*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)]];
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*n^2 - 16*n)*x + (40*n^2 - 112*n + 72)*x^2 + (13*n^2 - 39*n + 26)*x^3)*v[i - 1] - (64*n*x + (192*n - 256)*x^2 + (132*n - 256)*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)); 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