A292219 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 10\,x + 48\,x^{2} + 8\,n x - 56\,n x^{2} + 16\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-8\,x^{2} + 72\,x^{3} - 32\,n x^{3}\right)\,P_{n-1}'(x) + \left(16\,x^{4}\right)\,P_{n-1}''(x)\)

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

Rows ?

n	x⁰	x¹	x²	x³	x⁴	x⁵	x⁶	x⁷	x⁸	x⁹	x¹⁰	x¹¹	x¹²	x¹³	x¹⁴	P_n(1)
1	1															1
2	1	6														7
3	1	20	60													81
4	1	42	420	840												1303
5	1	72	1512	10080	15120											26785
6	1	110	3960	55440	277200	332640										669351
7	1	156	8580	205920	2162160	8648640	8648640									19674097
8	1	210	16380	600600	10810800	90810720	302702400	259459200								664400311
9	1	272	28560	1485120	40840800	588107520	4116752640	11762150400	8821612800							25330978113
10	1	342	46512	3255840	126977760	2793510720	33522128640	201132771840	502831929600	335221286400						1075631907655
11	1	420	71820	6511680	341863200	10666131840	195545750400	2011327718400	10559470521600	23465490048000	14079294028800					50322142646161
12	1	506	106260	12113640	823727520	34596555840	899510451840	14135164243200	127216478188800	593676898214400	1187353796428800	647647525324800				2570964805355607
13	1	600	151800	21252000	1817046000	98847302400	3459655584000	77100895872000	1060137318240000	8481098545920000	35620613892864000	64764752532480000	32382376266240000			142389639792952801
14	1	702	210600	35521200	3729726000	255113258400	11565134380800	346954031424000	6765603612768000	82690710822720000	595373117923584000	2273242813890048000	3788738023150080000	1748648318376960000		8495817365820681703
15	1	812	285012	57002400	7210803600	605707502400	34525327636800	1341555588172800	35215834189536000	610407792618624000	6714485718804864000	43949361068540928000	153822763739893248000	236650405753681920000	101421602465863680000	543205619067484203825

Technical details

Node id: n-1658117d1dedf548 · revision 0 · JSON
Definition source: derived:reversal
Fingerprint: f6-f136838b00e1c323

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 -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 = 1, …, ascending powers of x

%e Example

Triangle begins:
   1: 1;
   2: 1,   6;
   3: 1,  20,    60;
   4: 1,  42,   420,     840;
   5: 1,  72,  1512,   10080,     15120;
   6: 1, 110,  3960,   55440,    277200,     332640;
   7: 1, 156,  8580,  205920,   2162160,    8648640,     8648640;
   8: 1, 210, 16380,  600600,  10810800,   90810720,   302702400,    259459200;
   9: 1, 272, 28560, 1485120,  40840800,  588107520,  4116752640,  11762150400,   8821612800;
  10: 1, 342, 46512, 3255840, 126977760, 2793510720, 33522128640, 201132771840, 502831929600, 335221286400;

%F Formula

The row polynomials satisfy P(n,x) = (1+(8*n-10)*x+(16*n^2-56*n+48)*x^2)*P(n-1,x) - (8*x^2+(32*n-72)*x^3)*d/dx P(n-1,x) + 16*x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1.

%t Mathematica

P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (8*n - 10)*x + (16*n^2 - 56*n + 48)*x^2)*P[n - 1, x] - (8*x^2 + (32*n - 72)*x^3)*D[P[n - 1, x], x] + 16*x^4*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

%o PARI/GP

P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (1 + (8*n - 10)*x + (16*n^2 - 56*n + 48)*x^2)*v[i - 1] - (8*x^2 + (32*n - 72)*x^3)*deriv(v[i - 1], x) + 16*x^4*deriv(deriv(v[i - 1], x), x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Real-rooted? Every row P_n, n ≥ 1, 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 ≥ 1: the coefficients of P_n have one sign
derived from Affine, reversal and Möbius substitutions (elementary)
Hadamard kernel (Malo)? Rows P_n, n ≥ 1, 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. A generalization of unsigned Lah numbers, called L[4,3].?
  - Iterated real-rootedness preservers [SP14; Bra15, Thm 7.1 of arXiv:1410.6601; Bra06, Cor 3.5] polytool ? step › direct check
  - Row-hypergeometric rows (multiplier sequences) [SP14; Bra15, Thm 7.1 of arXiv:1410.6601] polytool RealRooted RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos RealRooted RealRooted.isPFMultiplierSequence_inv_ascPochhammer RealRooted RealRooted.IsPFMultiplierSequence.mul ? step › direct check

Related ?

Same fingerprint ?

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