Real-rooted atlas

A207606 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + 24\,x - 8\,n + 40\,n x + 4\,n^{2} + 16\,n^{2} x\right)\,P_{n}(x) = \left(-66 + 84\,x + 96\,x^{2} - 4\,n + 112\,n x + 80\,n x^{2} + 6\,n^{2} + 28\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-5 + 28\,x + 92\,x^{2} + 32\,x^{3} + 5\,n + 32\,n x + 52\,n x^{2} + 16\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 22 + 16\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
0221638
18813648272
23107445521281734
3102033363840185632010372
43216133442097615680560076859584
59848495609912010112055680157441792332864
6295221747284250645520004142401800964211240961821858
787080592712170025626883842567040151872054476810854492169816720
825361019517286453840120471681400784010536960513587215667202718722048052246090
97310646276912235164485066336069574400637109763944908816330752433152066560045056275295176
102089632198032648292624020265344032137376034757452826187033613759488049466880116019201599488983041438652672
115930668614914082846720647784336001401180480175030963215571691529950146564536576001440665603027763237847042129927466201148
1216730830188393632955564176289177286458282516808266187520849369292864066928643550288896142872576040617984077316096883916845875238519095006
13469543605706128083147201360104450561282331300672037036056960432116805123763921920024612268032120316416004331458560111437414419382272020414464983040197714750608
14131181406171126223210197878328368359324809022340880015879319948820755881484820528052019215497475840089360128000390578073601271626137629871063044782489604669440020971521010355299726
Technical details
Node id
n-f578f400880ba2bd · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8be6890b039ee0c2

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (affine: slope 1, intercept 1)
  • 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:     22,      16;
  1:     88,     136,       48;
  2:    310,     744,      552,      128;
  3:   1020,    3336,     3840,     1856,      320;
  4:   3216,   13344,    20976,    15680,     5600,      768;
  5:   9848,   49560,    99120,   101120,    55680,    15744,     1792;
  6:  29522,  174728,   425064,   552000,   414240,   180096,    42112,     4096;
  7:  87080,  592712,  1700256,  2688384,  2567040,  1518720,   544768,   108544,    9216;
  8: 253610, 1951728,  6453840, 12047168, 14007840, 10536960,  5135872,  1566720,  271872,  20480;
  9: 731064, 6276912, 23516448, 50663360, 69574400, 63710976, 39449088, 16330752, 4331520, 665600, 45056;
%F Formula
The row polynomials satisfy (4*n^2-8*n-12+(16*n^2+40*n+24)*x)*P(n,x) = (6*n^2-4*n-66+(28*n^2+112*n+84)*x+(16*n^2+80*n+96)*x^2)*P(n-1,x) + (5*n-5+(32*n+28)*x+(52*n+92)*x^2+(16*n+32)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 22 + 16*x.
%t Mathematica
P[0, x_] := 22 + 16*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n^2 - 4*n - 66 + (28*n^2 + 112*n + 84)*x + (16*n^2 + 80*n + 96)*x^2)*P[n - 1, x] + (5*n - 5 + (32*n + 28)*x + (52*n + 92)*x^2 + (16*n + 32)*x^3)*D[P[n - 1, x], x])/(4*n^2 - 8*n - 12 + (16*n^2 + 40*n + 24)*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 22 + 16*x; for(n = 1, N, my(i = n + 1); v[i] = ((6*n^2 - 4*n - 66 + (28*n^2 + 112*n + 84)*x + (16*n^2 + 80*n + 96)*x^2)*v[i - 1] + (5*n - 5 + (32*n + 28)*x + (52*n + 92)*x^2 + (16*n + 32)*x^3)*deriv(v[i - 1], x))/(4*n^2 - 8*n - 12 + (16*n^2 + 40*n + 24)*x)); 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

References

  1. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023