A075502 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 14\,x + 7\,n x\right)\,P_{n-1}(x) + \left(-7\,x^{2}\right)\,P_{n-1}'(x)\)

\(P_{0}(x) = 1 + 21\,x + 49\,x^{2}\)

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	x¹⁵	x¹⁶	P_n(1)
0	1	21	49															71
1	1	42	343	343														729
2	1	70	1225	5145	2401													8842
3	1	105	3185	30870	74431	16807												125399
4	1	147	6860	120050	722701	1058841	117649											2026249
5	1	196	13034	360150	4084101	16235562	14941423	823543										36458010
6	1	252	22638	907578	16689351	130590390	355888225	210003465	5764801									719866701
7	1	315	36750	2016840	54807627	714717675	4012419145	7683656190	2945813311	40353607								15453821461
8	1	385	56595	4074840	153632787	3016638009	29027537770	120031392250	164302593301	41281739961	282475249							358100141148
9	1	462	83545	7640325	381823827	10544644572	155726334148	1135995214200	3525181576301	3491636199282	578226834703	1977326743						8899677678109
10	1	546	119119	13488475	863164302	31926778884	672413918176	7676501248416	43285014073301	102196720335710	73902587019625	8097153012585	13841287201					235877034446341
11	1	637	164983	22660638	1807357552	86306129910	2460313535680	40624783239040	365698066506773	1617172212901245	2935410756419505	1560051480424710	113373983463391	96889010407				6634976621814472
12	1	735	222950	36519210	3552226678	212821158550	7897599720010	178402341237120	2356312445219733	16976491006185711	59536438207963080	83751552660170850	32874455072382301	1587332657497881	678223072849			197269776623577659
13	1	840	294980	56807660	6619840318	486342612756	22795080818510	675951123597750	12346843554498453	132435800821952628	772549060467762942	2167526889938878650	2377917929557166101	691950889177526202	22223335428043183	4747561509943		6177654735731310917
14	1	952	383180	85715700	11789337378	1042409199468	60243462000722	2271606780893450	54931764341156703	823859039873865996	7261903300743441714	34614587429584922214	78241359077417918851	67273652916778177030	14553192008156093425	311131443554114505	33232930569601	203136983117907790890

Technical details

Node id: n-8d8f552d6aeee89c · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-58d3614eea0b32aa

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,  21,    49;
  1: 1,  42,   343,     343;
  2: 1,  70,  1225,    5145,      2401;
  3: 1, 105,  3185,   30870,     74431,       16807;
  4: 1, 147,  6860,  120050,    722701,     1058841,       117649;
  5: 1, 196, 13034,  360150,   4084101,    16235562,     14941423,        823543;
  6: 1, 252, 22638,  907578,  16689351,   130590390,    355888225,     210003465,       5764801;
  7: 1, 315, 36750, 2016840,  54807627,   714717675,   4012419145,    7683656190,    2945813311,      40353607;
  8: 1, 385, 56595, 4074840, 153632787,  3016638009,  29027537770,  120031392250,  164302593301,   41281739961,    282475249;
  9: 1, 462, 83545, 7640325, 381823827, 10544644572, 155726334148, 1135995214200, 3525181576301, 3491636199282, 578226834703, 1977326743;

%F Formula

The row polynomials satisfy P(n,x) = (1+(7*n+14)*x)*P(n-1,x) - 7*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 21*x + 49*x^2.

%t Mathematica

P[0, x_] := 1 + 21*x + 49*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (7*n + 14)*x)*P[n - 1, x] - 7*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 + 21*x + 49*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (7*n + 14)*x)*v[i - 1] - 7*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 Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).?
  - 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 ?

Triangle read by rows: Stirling2 triangle with scaled diagonals (powers of 7).

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.