Real-rooted atlas

A088729 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 3\,x + 3\,n x + 2\,n x^{2} + 2\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-3\,x^{2} - 4\,n x^{3}\right)\,P_{n-1}'(x) + \left(2\,x^{4}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 1 + 9\,x + 13\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
1191323
21187975173
31302657655411602
414566538708311468317575
51631400136505910110098947293222497
618426183843027890196049813624395458353188806
71108449492610101243158445101670054520246445708726150988405
811357230199080305690726587575127331905310819590328972471102247563899222457
9116511055392040804734798334621709559950290230545061823145015805917469162263257317329515172
10119816225720225190482273112373883155560276194457819606939237790113110207877811064591139928091567595362164300173
11123423023125053541414802872107236118249468481023898075205504127238611741641891990295598169002522651919813255268583483818155216185781
1212733175920720708399591222154031903871925200044838970176033867247944531614122120110545872193348345706684325993104958979051663110641342970443196789115887252
13131542770330057016075859851911359501136544022901696669809120171394647554131147670024490994911939990321001266783600823650178653832427070111562012772619492302831909778535064722539020379
141360564205083260292923358113675201643047553723505699743141950742257741040536647227809342123996715082570574155207137332510903663570149543850147637852615465858286059507457979595315654681981355138457553073641465
1514087310076041005117073782349873817275721530427417343258748530282974805959703326690437187228426269180181842962143298298817995230509249312448439291110809481989077147013366034601755147057486055905423594051303707670291359014006059432756066914
Technical details
Node id
n-7d867fcc3c35d8d0 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ad16784f8d87fb89

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 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,   9,    13;
   2: 1,  18,    79,     75;
   3: 1,  30,   265,    765,      541;
   4: 1,  45,   665,   3870,     8311,      4683;
   5: 1,  63,  1400,  13650,    59101,    100989,      47293;
   6: 1,  84,  2618,  38430,   278901,    960498,    1362439,      545835;
   7: 1, 108,  4494,  92610,  1012431,   5844510,   16700545,    20246445,     7087261;
   8: 1, 135,  7230, 199080,  3056907,  26587575,  127331905,   310819590,   328972471,    102247563;
   9: 1, 165, 11055, 392040,  8047347,  98334621,  709559950,  2902305450,  6182314501,   5805917469,   1622632573;
  10: 1, 198, 16225, 720225, 19048227, 311237388, 3155560276, 19445781960, 69392377901, 131102078778, 110645911399, 28091567595;
%F Formula
The row polynomials satisfy P(n,x) = (1+(3*n+3)*x+(2*n^2+2*n)*x^2)*P(n-1,x) - (3*x^2+4*n*x^3)*d/dx P(n-1,x) + 2*x^4*d^2/dx^2 P(n-1,x) for n >= 2, with P(1,x) = 1 + 9*x + 13*x^2.
%t Mathematica
P[1, x_] := 1 + 9*x + 13*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (3*n + 3)*x + (2*n^2 + 2*n)*x^2)*P[n - 1, x] - (3*x^2 + 4*n*x^3)*D[P[n - 1, x], x] + 2*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 + 9*x + 13*x^2; for(n = 2, N, my(i = n); v[i] = (1 + (3*n + 3)*x + (2*n^2 + 2*n)*x^2)*v[i - 1] - (3*x^2 + 4*n*x^3)*deriv(v[i - 1], x) + 2*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

Proof chains ? JSON

Same fingerprint ?

References

  1. [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
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.