A269947 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-64 + 16\,x - x^{2} + 192\,n - 24\,n x - 240\,n^{2} + 12\,n^{2} x + 160\,n^{3} - 2\,n^{3} x - 60\,n^{4} + 12\,n^{5} - n^{6}\right)\,P_{n-2}(x) + \left(-9 + 2\,x + 15\,n - 9\,n^{2} + 2\,n^{3}\right)\,P_{n-1}(x)\)

\(P_{0}(x) = 1\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
0	1															1
1	1	1														2
2	1	3	1													5
3	8	26	12	1												47
4	216	718	359	40	1											1334
5	13824	46384	23945	2955	105	1										87214
6	1728000	5825648	3055789	395875	16180	231	1									11021724
7	373248000	1261795968	667924896	88900444	3905810	66301	448	1								2395841868
8	128024064000	433542513024	230804309280	31235327492	1432180809	26710708	220406	792	1							825065326512
9	65548320768000	222229814796288	118758108088896	16249307013440	765816976509	15133484505	139773498	626694	1305	1						423566641529136
10	47784725839872000	162136631628029952	86875231358615616	11977975374578240	575224096218965	11812447909863	117163823763	597249750	1579335	2035	1					309361718507879520
11	47784725839872000000	162232201079709696000	87094543943137724928	12074750365425034496	587721625565143080	12398806031005790	129089342170536	714998334633	2178145860	3616360	3036	1				309786471466262874720
12	63601470092869632000000	216026629088773349376000	116142293927015940423168	16168544215764931598080	794861687723718492856	17102186031729274410	184340924982586031	1081449956934732	3616256345163	6995107130	7660301	4368	1			412751086624177499802240
13	109903340320478724096000000	373421218005586086985728000	200986123114470270259298304	28068696603578505769150464	1390404200453869713341504	30363479517792803154256	335820274200109734893	2054139468638772515	7333894906872039	15710720449083	20239719643	15212561	6097	1		713800483605848544257439360
14	241457638684091756838912000000	820626222638913590555836416000	442071692639339323528108965888	61890988289267086308021012480	3084034967128965176688186176	68127401115488450044359952	768482466114434057720217	4850761271828056197327	18173900884046082890	41865857722110873	60197457689314	53676855713	28613754	8295	1	1569199477971299424112656218880

Technical details

Node id: n-caf0daa742bb1adc · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-fe234fa46304cfff

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:              1,               1;
  2:              1,               3,               1;
  3:              8,              26,              12,              1;
  4:            216,             718,             359,             40,            1;
  5:          13824,           46384,           23945,           2955,          105,           1;
  6:        1728000,         5825648,         3055789,         395875,        16180,         231,         1;
  7:      373248000,      1261795968,       667924896,       88900444,      3905810,       66301,       448,      1;
  8:   128024064000,    433542513024,    230804309280,    31235327492,   1432180809,    26710708,    220406,    792,    1;
  9: 65548320768000, 222229814796288, 118758108088896, 16249307013440, 765816976509, 15133484505, 139773498, 626694, 1305, 1;

%F Formula

The row polynomials satisfy P(n,x) = -(n^6-12*n^5+60*n^4-160*n^3+240*n^2-192*n+64+(2*n^3-12*n^2+24*n-16)*x+x^2)*P(n-2,x) + (2*n^3-9*n^2+15*n-9+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[-(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*P[n - 2, x] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*P[n - 1, x]];
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] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = -(n^6 - 12*n^5 + 60*n^4 - 160*n^3 + 240*n^2 - 192*n + 64 + (2*n^3 - 12*n^2 + 24*n - 16)*x + x^2)*v[i - 2] + (2*n^3 - 9*n^2 + 15*n - 9 + 2*x)*v[i - 1]); 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 Triangle read by rows, Stirling cycle numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-…?
  - 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
  - Parent-row provers of polytool polytool ? step › direct check

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.