Real-rooted atlas

A121574 → basis A013609 → reverse

nonnegative coefficients

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

Recurrence

\(\left(4\,n\right)\,P_{n}(x) = \left(36\,x^{2} + 26\,x^{3} + 8\,n - 4\,n x - 56\,n x^{2} - 39\,n x^{3} + 16\,n^{2} x + 20\,n^{2} x^{2} + 13\,n^{2} x^{3}\right)\,P_{n-1}(x) + \left(64\,x^{2} + 128\,x^{3} + 52\,x^{4} - 8\,n x - 48\,n x^{2} - 66\,n x^{3} - 26\,n x^{4}\right)\,P_{n-1}'(x) + \left(8\,x^{2} + 44\,x^{3} + 46\,x^{4} + 13\,x^{5}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1235
24141028
38447033155
416120304310109859
5323041064171612843604760
6647363296735286405100118926377
712817289440270244383240404196823927146165
8256396825600896961867362369361794167437812970809956
951289606668827680070604811487681194280766772276610428374488275
1010241996816844880883224474884878848649547257121523181708101582214148124871243
112048440324152322265344794176018789056306310723444804826227696128967603692712467280137821040
1240969625610035206133248244656646710720012983449617865910417371931211655217651289056133114801543321763718929
138192208896238592016151552722769922257602565065104648268728969829431048411657285042558562007723124764835450972434232057765
14163844505605595136415641602062786567234160641848981760350005337649336576005157258496393883256021334424807755256601695376701683505023451445612
Technical details
Node id
n-9809095053fda73e · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-a319b0b1324ceacf

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,    3;
  2:   4,   14,    10;
  3:   8,   44,    70,     33;
  4:  16,  120,   304,    310,    109;
  5:  32,  304,  1064,   1716,   1284,     360;
  6:  64,  736,  3296,   7352,   8640,    5100,    1189;
  7: 128, 1728,  9440,  27024,  43832,   40404,   19682,   3927;
  8: 256, 3968, 25600,  89696, 186736,  236936,  179416,  74378,  12970;
  9: 512, 8960, 66688, 276800, 706048, 1148768, 1194280, 766772, 276610, 42837;
%F Formula
The row polynomials satisfy 4*n*P(n,x) = (8*n+(16*n^2-4*n)*x+(20*n^2-56*n+36)*x^2+(13*n^2-39*n+26)*x^3)*P(n-1,x) - (8*n*x+(48*n-64)*x^2+(66*n-128)*x^3+(26*n-52)*x^4)*d/dx P(n-1,x) + (8*x^2+44*x^3+46*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[((8*n + (16*n^2 - 4*n)*x + (20*n^2 - 56*n + 36)*x^2 + (13*n^2 - 39*n + 26)*x^3)*P[n - 1, x] - (8*n*x + (48*n - 64)*x^2 + (66*n - 128)*x^3 + (26*n - 52)*x^4)*D[P[n - 1, x], x] + (8*x^2 + 44*x^3 + 46*x^4 + 13*x^5)*D[P[n - 1, x], {x, 2}])/(4*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] = ((8*n + (16*n^2 - 4*n)*x + (20*n^2 - 56*n + 36)*x^2 + (13*n^2 - 39*n + 26)*x^3)*v[i - 1] - (8*n*x + (48*n - 64)*x^2 + (66*n - 128)*x^3 + (26*n - 52)*x^4)*deriv(v[i - 1], x) + (8*x^2 + 44*x^3 + 46*x^4 + 13*x^5)*deriv(deriv(v[i - 1], x), x))/(4*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