A257614 → D

nonnegative coefficients

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

Recurrence

\(\left(-8 + 5\,n\right)\,P_{n}(x) = \left(42 - 156\,x + 312\,x^{2} - 70\,n + 320\,n x - 835\,n x^{2} - 100\,n^{2} x + 600\,n^{2} x^{2} - 125\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(30\,x - 150\,x^{2} + 120\,x^{3} - 50\,n x + 325\,n x^{2} - 275\,n x^{3} - 125\,n^{2} x^{2} + 125\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-62 + 128\,x + 45\,n - 160\,n x + 50\,n^{2} x\right)\,P_{n-1}(x) + \left(-40\,x + 40\,x^{2} + 25\,n x - 25\,n x^{2}\right)\,P_{n-1}'(x)\)

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

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

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	28	8													36
3	244	488	24												756
4	1844	11712	5532	64											19152
5	13260	203240	304860	53040	160										574560
6	93684	3022320	10365240	6044640	468420	384									19994688
7	657836	41326776	275945640	367927520	103316940	3947016	896								793122624
8	4609588	538022816	6364810308	16188810880	10608017180	1614068448	32267116	2048							35350608384
9	32277868	6797383304	134026870404	585287010448	731608763060	268053740808	23790841564	258222944	4608						1749855115008
10	225969140	84279940560	2655711570240	18594794033760	39506873205240	27892191050640	6196660330560	337119762240	2033722260	10240					95269889594880
11	1581837228	1032600385880	50456732949360	540100572874560	1810477337891880	2172572805470256	945176002530480	134551287864960	4646701736460	15818372280	22528				5659031441935872
12	11072977332	12555715702272	930562787343660	14707789648404480	73862539618529160	139044659550096384	103407555465940824	29415579296808960	2791688362030980	62778578511360	121802750652	49152			364235841899145216
13	77511095276	151930907843112	16798913209619436	381886640271767920	2766447526241522520	7728925864665778992	9017080175443408824	4426316041986436032	859244940611477820	55996377365398120	835619993137116	930133143312	106496		25253685038340734976
14	542578215796	1832782269931568	298571617184116488	9566230735179175136	97129923324487473340	386754582991877662752	667263932982812252976	515672777322503550336	174833861984077452012	23915576837947937840	1094762596341760456	10996693619589408	7053516805348	229376	1876543057464396152832

Technical details

Node id: n-ba4ac7bddd5cc4ce · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-4f56baeab55bc9ab

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): no

Matches ?

No OEIS matches.

Submission text ?

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

%e Example

Triangle begins:
  0:        0;
  1:        2;
  2:       28,          8;
  3:      244,        488,           24;
  4:     1844,      11712,         5532,           64;
  5:    13260,     203240,       304860,        53040,          160;
  6:    93684,    3022320,     10365240,      6044640,       468420,          384;
  7:   657836,   41326776,    275945640,    367927520,    103316940,      3947016,         896;
  8:  4609588,  538022816,   6364810308,  16188810880,  10608017180,   1614068448,    32267116,      2048;
  9: 32277868, 6797383304, 134026870404, 585287010448, 731608763060, 268053740808, 23790841564, 258222944, 4608;

%F Formula

The row polynomials satisfy (5*n-8)*P(n,x) = -(70*n-42+(100*n^2-320*n+156)*x+(125*n^3-600*n^2+835*n-312)*x^2)*P(n-2,x) - ((50*n-30)*x+(125*n^2-325*n+150)*x^2-(125*n^2-275*n+120)*x^3)*d/dx P(n-2,x) + (45*n-62+(50*n^2-160*n+128)*x)*P(n-1,x) + ((25*n-40)*x-(25*n-40)*x^2)*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[(-(70*n - 42 + (100*n^2 - 320*n + 156)*x + (125*n^3 - 600*n^2 + 835*n - 312)*x^2)*P[n - 2, x] - ((50*n - 30)*x + (125*n^2 - 325*n + 150)*x^2 - (125*n^2 - 275*n + 120)*x^3)*D[P[n - 2, x], x] + (45*n - 62 + (50*n^2 - 160*n + 128)*x)*P[n - 1, x] + ((25*n - 40)*x - (25*n - 40)*x^2)*D[P[n - 1, x], x])/(5*n - 8)]];
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] = (-(70*n - 42 + (100*n^2 - 320*n + 156)*x + (125*n^3 - 600*n^2 + 835*n - 312)*x^2)*v[i - 2] - ((50*n - 30)*x + (125*n^2 - 325*n + 150)*x^2 - (125*n^2 - 275*n + 120)*x^3)*deriv(v[i - 2], x) + (45*n - 62 + (50*n^2 - 160*n + 128)*x)*v[i - 1] + ((25*n - 40)*x - (25*n - 40)*x^2)*deriv(v[i - 1], x))/(5*n - 8)); 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
Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, unordered) for every m ≥ 1
by Laguerre’s theorem on differential operators

Proof chains ? JSON

Laguerre’s theorem on differential operators [SP14] polytool ? step ›
- from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Affine, reversal and Möbius substitutions (elementary) polytool ? step ›
    
    from Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m)…?
    
    ↑ Liu–Wang theorem (above)

References

[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
[SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.