Real-rooted atlas

A249252 → basis A007318 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

\(\left(-2 + 2\,n\right)\,P_{n}(x) = \left(-2 - 2\,n + 4\,n^{2}\right)\,P_{n-1}(x) + \left(84 + 66\,x + 12\,x^{2} - 41\,n - 28\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x) + \left(82 + 97\,x + 36\,x^{2} + 4\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 9 + 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
19413
24940897
328133212816757
4160124801416352325881
59129174921321649128966445709
65204911884011168855232151362176128355249
72967617865248839685456161995524320051202562760997
8169200151056006677320494032022600006540801168641177651221458473
996470493265162048706720420066402307008083623041999360303872266241024166775293
1055003249206358600345757320340562560218284320950880002849702457989127664645939220481296177889
11313604441129170007624016339202659767440194884032099159801635738316891202816161443841887232131072409610073896981
1217880352018021150160163873162322015586992016617998880967259136040742759681250970112277791232434954244556800286720819278294346201
13101945937694947583038811016881731214899631966413651663040089463027264431292912641548298009641378795528128972801140736001082572862259216384608503805293
14581251096493034290105207313873125841078717345088108735952668879227476032043021479744017682591539255268463104130698096642301380608292536320253706241343488327684729292714257
Technical details
Node id
n-5b44a84154f4b80a · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-9dd229496f4f95ab

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:       9,        4;
  2:      49,       40,        8;
  3:     281,      332,      128,       16;
  4:    1601,     2480,     1416,      352,       32;
  5:    9129,    17492,    13216,     4912,      896,      64;
  6:   52049,   118840,   111688,    55232,    15136,    2176,     128;
  7:  296761,   786524,   883968,   545616,   199552,   43200,    5120,    256;
  8: 1692001,  5105600,  6677320,  4940320,  2260000,  654080,  116864,  11776,   512;
  9: 9647049, 32651620, 48706720, 42006640, 23070080, 8362304, 1999360, 303872, 26624, 1024;
%F Formula
The row polynomials satisfy (2*n-2)*P(n,x) = (4*n^2-2*n-2)*P(n-1,x) - (41*n-84+(28*n-66)*x+(4*n-12)*x^2)*d/dx P(n-1,x) + (82+97*x+36*x^2+4*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 9 + 4*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 9 + 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 - 2*n - 2)*P[n - 1, x] - (41*n - 84 + (28*n - 66)*x + (4*n - 12)*x^2)*D[P[n - 1, x], x] + (82 + 97*x + 36*x^2 + 4*x^3)*D[P[n - 1, x], {x, 2}])/(2*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 9 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*n^2 - 2*n - 2)*v[i - 1] - (41*n - 84 + (28*n - 66)*x + (4*n - 12)*x^2)*deriv(v[i - 1], x) + (82 + 97*x + 36*x^2 + 4*x^3)*deriv(deriv(v[i - 1], x), x))/(2*n - 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

Candidate equalities ?

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