A075500 → reverse

nonnegative coefficients

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

Recurrence

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

\(P_{0}(x) = 1 + 15\,x + 25\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	15	25															41
1	1	30	175	125														331
2	1	50	625	1875	625													3176
3	1	75	1625	11250	19375	3125												35451
4	1	105	3500	43750	188125	196875	15625											447981
5	1	140	6650	131250	1063125	3018750	1984375	78125										6282416
6	1	180	11550	330750	4344375	24281250	47265625	19921875	390625									96546231
7	1	225	18750	735000	14266875	132890625	532890625	728906250	199609375	1953125								1611270851
8	1	275	28875	1485000	39991875	560896875	3855156250	11386718750	11133203125	1998046875	9765625							28985293526
9	1	330	42625	2784375	99391875	1960612500	20682062500	107765625000	238867578125	168996093750	19990234375	48828125						558413253581
10	1	390	60775	4915625	224688750	5936287500	89303500000	728227500000	2933008203125	4946347656250	2554931640625	199951171875	244140625					11458179765541
11	1	455	84175	8258250	470470000	16047281250	326755000000	3853850000000	24779833203125	78271552734375	101481884765625	38523925781250	1999755859375	1220703125				249255304141006
12	1	525	113750	13308750	924673750	39570781250	1048882656250	16924050000000	159664583203125	821666548828125	2058270703125000	2068161621093750	579858642578125	19998779296875	6103515625			5725640423174901
13	1	600	150500	20702500	1723198750	90427837500	3027421718750	64123769531250	836626583203125	6409926960937500	26708267167968750	53524929199218750	41943091064453125	8717878417968750	199993896484375	30517578125		138407987170952351
14	1	680	195500	31237500	3068861250	193819762500	8000952781250	215494855468750	3722196212109375	39874990289062500	251055710800781250	854772944238281250	1380066321044921875	847579699707031250	130968170166015625	1999969482421875	152587890625	3510263847256823056

Technical details

Node id: n-7f851af1bfd10b81 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-9e37acaa9ec1379f

Observed on the stored rows ?

Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
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,  15,    25;
  1: 1,  30,   175,     125;
  2: 1,  50,   625,    1875,      625;
  3: 1,  75,  1625,   11250,    19375,       3125;
  4: 1, 105,  3500,   43750,   188125,     196875,       15625;
  5: 1, 140,  6650,  131250,  1063125,    3018750,     1984375,        78125;
  6: 1, 180, 11550,  330750,  4344375,   24281250,    47265625,     19921875,       390625;
  7: 1, 225, 18750,  735000, 14266875,  132890625,   532890625,    728906250,    199609375,      1953125;
  8: 1, 275, 28875, 1485000, 39991875,  560896875,  3855156250,  11386718750,  11133203125,   1998046875,     9765625;
  9: 1, 330, 42625, 2784375, 99391875, 1960612500, 20682062500, 107765625000, 238867578125, 168996093750, 19990234375, 48828125;

%F Formula

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

%t Mathematica

P[0, x_] := 1 + 15*x + 25*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (5*n + 10)*x)*P[n - 1, x] - 5*x^2*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] = 1 + 15*x + 25*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (5*n + 10)*x)*v[i - 1] - 5*x^2*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 Affine, reversal and Möbius substitutions (elementary)
Coefficients of one sign? Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Affine, reversal and Möbius substitutions (elementary)

Proof chains ? JSON

Affine, reversal and Möbius substitutions (elementary) 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

Related ?

Same fingerprint ?

Stirling2 triangle with scaled diagonals (powers of 5).

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.