A103633 → basis A013609 → D

nonnegative coefficientsonly rational roots

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

Recurrence

\(\left(14 + 44\,x + 32\,x^{2} - 23\,n - 68\,n x - 48\,n x^{2} + 9\,n^{2} + 24\,n^{2} x + 16\,n^{2} x^{2}\right)\,P_{n}(x) = \left(8 + 48\,x + 96\,x^{2} + 64\,x^{3} - 112\,n - 486\,n x - 712\,n x^{2} - 352\,n x^{3} + 111\,n^{2} + 454\,n^{2} x + 624\,n^{2} x^{2} + 288\,n^{2} x^{3} - 27\,n^{3} - 108\,n^{3} x - 144\,n^{3} x^{2} - 64\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(4 + 28\,x + 72\,x^{2} + 80\,x^{3} + 32\,x^{4} - 30\,n - 178\,n x - 388\,n x^{2} - 368\,n x^{3} - 128\,n x^{4} + 18\,n^{2} + 102\,n^{2} x + 212\,n^{2} x^{2} + 192\,n^{2} x^{3} + 64\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 0\)

\(P_{1}(x) = 2\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	P_n(1)
0															0
1	2														2
2	6	8													14
3	10	32	24												66
4	24	104	144	64											336
5	32	200	456	448	160										1296
6	72	528	1512	2112	1440	384									6048
7	88	816	3096	6144	6720	3840	896								21600
8	192	1984	8640	20544	28800	23808	10752	2048							96768
9	224	2752	14592	43584	80160	92928	66304	26624	4608						331776
10	480	6400	37440	126080	269280	378240	349440	204800	69120	10240					1451520
11	544	8320	56640	225920	584480	1024512	1232000	1003520	529920	163840	22528				4852224
12	1152	18816	138240	602880	1733760	3451776	4854528	4823040	3317760	1505280	405504	49152			20901888
13	1280	23424	194688	971520	3240960	7612800	12908672	15919104	14169600	8878080	3717120	933888	106496		68677632
14	2688	51968	459648	2462208	8910720	23003904	43580544	61343744	64157184	49244160	26966016	9977856	2236416	229376	292626432

Technical details

Node id: n-68ae4e0393bd3f70 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-1788d34a8315ad53

Observed on the stored rows ?

Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x

%e Example

Triangle begins:
  0:   0;
  1:   2;
  2:   6,    8;
  3:  10,   32,    24;
  4:  24,  104,   144,    64;
  5:  32,  200,   456,   448,   160;
  6:  72,  528,  1512,  2112,  1440,   384;
  7:  88,  816,  3096,  6144,  6720,  3840,   896;
  8: 192, 1984,  8640, 20544, 28800, 23808, 10752,  2048;
  9: 224, 2752, 14592, 43584, 80160, 92928, 66304, 26624, 4608;

%F Formula

The row polynomials satisfy (9*n^2-23*n+14+(24*n^2-68*n+44)*x+(16*n^2-48*n+32)*x^2)*P(n,x) = -(27*n^3-111*n^2+112*n-8+(108*n^3-454*n^2+486*n-48)*x+(144*n^3-624*n^2+712*n-96)*x^2+(64*n^3-288*n^2+352*n-64)*x^3)*P(n-1,x) + (18*n^2-30*n+4+(102*n^2-178*n+28)*x+(212*n^2-388*n+72)*x^2+(192*n^2-368*n+80)*x^3+(64*n^2-128*n+32)*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.

%t Mathematica

P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(27*n^3 - 111*n^2 + 112*n - 8 + (108*n^3 - 454*n^2 + 486*n - 48)*x + (144*n^3 - 624*n^2 + 712*n - 96)*x^2 + (64*n^3 - 288*n^2 + 352*n - 64)*x^3)*P[n - 1, x] + (18*n^2 - 30*n + 4 + (102*n^2 - 178*n + 28)*x + (212*n^2 - 388*n + 72)*x^2 + (192*n^2 - 368*n + 80)*x^3 + (64*n^2 - 128*n + 32)*x^4)*D[P[n - 1, x], x])/(9*n^2 - 23*n + 14 + (24*n^2 - 68*n + 44)*x + (16*n^2 - 48*n + 32)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(27*n^3 - 111*n^2 + 112*n - 8 + (108*n^3 - 454*n^2 + 486*n - 48)*x + (144*n^3 - 624*n^2 + 712*n - 96)*x^2 + (64*n^3 - 288*n^2 + 352*n - 64)*x^3)*v[i - 1] + (18*n^2 - 30*n + 4 + (102*n^2 - 178*n + 28)*x + (212*n^2 - 388*n + 72)*x^2 + (192*n^2 - 368*n + 80)*x^3 + (64*n^2 - 128*n + 32)*x^4)*deriv(v[i - 1], x))/(9*n^2 - 23*n + 14 + (24*n^2 - 68*n + 44)*x + (16*n^2 - 48*n + 32)*x^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

Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0]
by Laguerre’s theorem on differential operators
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Laguerre’s theorem on differential operators
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from basis_map_A013609 of Triangle read by rows: triangle of repeated stepped binomial coefficients.?
  - Geometric basis map (elementary) polytool ? step ›
    
    from Triangle read by rows: triangle of repeated stepped binomial coefficients.?
    
    Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
    
    Parent-row provers of polytool polytool ? step › direct check

Related ?

Same fingerprint ?

A064861 → D

References

[Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
[Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.