A257616 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-8 + 6\,x + 3\,n\right)\,P_{n}(x) = \left(80 - 476\,x + 1616\,x^{2} - 1692\,x^{3} + 360\,x^{4} - 48\,n + 348\,n x - 1800\,n x^{2} + 1512\,n x^{3} - 216\,n x^{4} - 72\,n^{2} x + 756\,n^{2} x^{2} - 324\,n^{2} x^{3} - 108\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(60\,x - 612\,x^{2} + 1308\,x^{3} - 972\,x^{4} + 216\,x^{5} - 36\,n x + 504\,n x^{2} - 792\,n x^{3} + 324\,n x^{4} - 108\,n^{2} x^{2} + 108\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-74 + 268\,x - 222\,x^{2} + 36\,x^{3} + 30\,n - 174\,n x + 90\,n x^{2} + 36\,n^{2} x\right)\,P_{n-1}(x) + \left(-48\,x + 84\,x^{2} - 36\,x^{3} + 18\,n x - 18\,n x^{2}\right)\,P_{n-1}'(x)\)

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

\(P_{1}(x) = 4 + 2\,x\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	P_n(1)
0	1								1
1	4	2							6
2	36	40	4						80
3	320	936	336	8					1600
4	2672	20128	16704	2720	16				42240
5	21696	372512	701696	262080	21824	32			1379840
6	174400	6212736	24070080	19093760	3890880	174720	64		53616640
7	1397248	97092224	704362752	1094192000	458076160	56230272	1398016	128	2412748800

All 15 rows

Technical details

Node id: n-401b73bdd722880d · revision 0 · JSON
Definition source: derived:one_plus_derivative
Fingerprint: f6-2b6b8baf2ca9e23d

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:        4,           2;
  2:       36,          40,            4;
  3:      320,         936,          336,             8;
  4:     2672,       20128,        16704,          2720,            16;
  5:    21696,      372512,       701696,        262080,         21824,            32;
  6:   174400,     6212736,     24070080,      19093760,       3890880,        174720,           64;
  7:  1397248,    97092224,    704362752,    1094192000,     458076160,      56230272,      1398016,         128;
  8: 11182848,  1457095168,  18446625280,   51426276864,   40919045120,   10245627392,    801234432,    11184640,      256;
  9: 89474048, 21316317696, 447503818752, 2085318076416, 2909531916288, 1358702779392, 219924086784, 11329210368, 89478144, 512;

%F Formula

The row polynomials satisfy (3*n-8+6*x)*P(n,x) = -(48*n-80+(72*n^2-348*n+476)*x+(108*n^3-756*n^2+1800*n-1616)*x^2+(324*n^2-1512*n+1692)*x^3+(216*n-360)*x^4)*P(n-2,x) - ((36*n-60)*x+(108*n^2-504*n+612)*x^2-(108*n^2-792*n+1308)*x^3-(324*n-972)*x^4-216*x^5)*d/dx P(n-2,x) + (30*n-74+(36*n^2-174*n+268)*x+(90*n-222)*x^2+36*x^3)*P(n-1,x) + ((18*n-48)*x-(18*n-84)*x^2-36*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 2*x.

%t Mathematica

P[0, x_] := 1;
P[1, x_] := 4 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(48*n - 80 + (72*n^2 - 348*n + 476)*x + (108*n^3 - 756*n^2 + 1800*n - 1616)*x^2 + (324*n^2 - 1512*n + 1692)*x^3 + (216*n - 360)*x^4)*P[n - 2, x] - ((36*n - 60)*x + (108*n^2 - 504*n + 612)*x^2 - (108*n^2 - 792*n + 1308)*x^3 - (324*n - 972)*x^4 - 216*x^5)*D[P[n - 2, x], x] + (30*n - 74 + (36*n^2 - 174*n + 268)*x + (90*n - 222)*x^2 + 36*x^3)*P[n - 1, x] + ((18*n - 48)*x - (18*n - 84)*x^2 - 36*x^3)*D[P[n - 1, x], x])/(3*n - 8 + 6*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] = 4 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(48*n - 80 + (72*n^2 - 348*n + 476)*x + (108*n^3 - 756*n^2 + 1800*n - 1616)*x^2 + (324*n^2 - 1512*n + 1692)*x^3 + (216*n - 360)*x^4)*v[i - 2] - ((36*n - 60)*x + (108*n^2 - 504*n + 612)*x^2 - (108*n^2 - 792*n + 1308)*x^3 - (324*n - 972)*x^4 - 216*x^5)*deriv(v[i - 2], x) + (30*n - 74 + (36*n^2 - 174*n + 268)*x + (90*n - 222)*x^2 + 36*x^3)*v[i - 1] + ((18*n - 48)*x - (18*n - 84)*x^2 - 36*x^3)*deriv(v[i - 1], x))/(3*n - 8 + 6*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

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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.?
  - 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); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 6*x + 2.?
    
    ↑ Liu–Wang theorem (above)

Related ?

Same fingerprint ?

A142458 → D+1

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.