A075500 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-50 - 10\,x - x^{2}\right)\,P_{n-2}(x) + \left(-25\,x - 5\,x^{2}\right)\,P_{n-2}'(x) + \left(15 + 2\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 15 + 2\,x\)

\(P_{1}(x) = 175 + 60\,x + 3\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	P_n(1)
0	15	2															17
1	175	60	3														238
2	1875	1250	150	4													3279
3	19375	22500	4875	300	5												47055
4	196875	376250	131250	14000	525	6											718906
5	1984375	6037500	3189375	525000	33250	840	7										11770347
6	19921875	94531250	72843750	17377500	1653750	69300	1260	8									206398693
7	199609375	1457812500	1598671875	531562500	71334375	4410000	131250	1800	9								3863533684
8	1998046875	22266406250	34160156250	15420625000	2804484375	239951250	10395000	231000	2475	10							76900298485
9	19990234375	337992187500	716602734375	431062500000	103410312500	11763675000	695743125	22275000	383625	3300	11						1621540048811
10	199951171875	5109863281250	14839042968750	11732032812500	3641137500000	535821000000	41554012500	1797510000	44240625	607750	4290	12					36101245109552
11	1999755859375	77047851562500	304445654296875	313086210937500	123899166015625	23123100000000	2287285000000	128378250000	4234230000	82582500	925925	5460	13				846021719665773
12	19998779296875	1159717285156250	6204484863281250	8233082812500000	4108332744140625	957987499218750	118468350000000	8391061250000	356137031250	9246737500	146396250	1365000	6825	14			20810828926380589
13	199993896484375	17435756835937500	125829273193359375	214099716796875000	133541335839843750	38459561765625000	5856386082421875	512990156250000	27246795468750	904278375000	18955186250	248430000	1956500	8400	15		535963184846221790
14	1999969482421875	261936340332031250	2542739099121093750	5520265284179687500	4273864721191406250	1506334264804687500	279124932023437500	29777569696875000	1939453699218750	80009527812500	2132017387500	36826335000	406087500	2737000	10200	16	14418063813311229091

Technical details

Node id: n-6afa27ca0f35db12 · revision 0 · JSON
Definition source: derived:derivative
Fingerprint: f6-76477e71dc51917c

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:          15,            2;
  1:         175,           60,            3;
  2:        1875,         1250,          150,            4;
  3:       19375,        22500,         4875,          300,            5;
  4:      196875,       376250,       131250,        14000,          525,           6;
  5:     1984375,      6037500,      3189375,       525000,        33250,         840,         7;
  6:    19921875,     94531250,     72843750,     17377500,      1653750,       69300,      1260,        8;
  7:   199609375,   1457812500,   1598671875,    531562500,     71334375,     4410000,    131250,     1800,      9;
  8:  1998046875,  22266406250,  34160156250,  15420625000,   2804484375,   239951250,  10395000,   231000,   2475,   10;
  9: 19990234375, 337992187500, 716602734375, 431062500000, 103410312500, 11763675000, 695743125, 22275000, 383625, 3300, 11;

%F Formula

The row polynomials satisfy P(n,x) = -(50+10*x+x^2)*P(n-2,x) - (25*x+5*x^2)*d/dx P(n-2,x) + (15+2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 15 + 2*x, P(1,x) = 175 + 60*x + 3*x^2.

%t Mathematica

P[0, x_] := 15 + 2*x;
P[1, x_] := 175 + 60*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(50 + 10*x + x^2)*P[n - 2, x] - (25*x + 5*x^2)*D[P[n - 2, x], x] + (15 + 2*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N + 1)); v[1] = 15 + 2*x; v[2] = 175 + 60*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(50 + 10*x + x^2)*v[i - 2] - (25*x + 5*x^2)*deriv(v[i - 2], x) + (15 + 2*x)*v[i - 1] + 5*x*deriv(v[i - 1], 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 Stirling2 triangle with scaled diagonals (powers of 5).?
  - Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ? step › direct check
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] 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
[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.