Real-rooted atlas

A322944 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 - 7\,x + 4\,n x\right)\,P_{n-1}(x) + \left(-3\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
31629
411538660
512818827224513
6145580234022001205286
7166139011040302801992072063417
81912842374502069204135601999205040865824
9112052081028169559203931200598976022041604032013229505
1011538808243432342350424005520767513609205056026490240362880223336458
11119014010516480102396001095931206048728001550326400149929920034473600036288004123226601
12123121230100683026766960406541520345429392015462400800325058272002583282240048299328003991680082559530692
13127630932183480063012840128985120015650539520108728336640403603446400708455193600469820736000724889088004790016001780580892609
1413254362831648761364048403621326280595054803205938950617603447996872320107985413536001605583499520089990828928001160301542400622702080041125146159150
151378598785215392275659384921392472019711587896026765868729602245263884864011079045065088029681791018880037828328778240018114195939840019731353241600871782912001012187976013593
Technical details
Node id
n-307f8a9151c09013 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-4e1886c021cf9741

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,   1;
   3: 1,   6,    2;
   4: 1,  15,   38,      6;
   5: 1,  28,  188,    272,      24;
   6: 1,  45,  580,   2340,    2200,      120;
   7: 1,  66, 1390,  11040,   30280,    19920,      720;
   8: 1,  91, 2842,  37450,  206920,   413560,   199920,     5040;
   9: 1, 120, 5208, 102816,  955920,  3931200,  5989760,  2204160,    40320;
  10: 1, 153, 8808, 243432, 3423504, 24005520, 76751360, 92050560, 26490240, 362880;
%F Formula
The row polynomials satisfy P(n,x) = (1+(4*n-7)*x)*P(n-1,x) - 3*x^2*d/dx 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 + (4*n - 7)*x)*P[n - 1, x] - 3*x^2*D[P[n - 1, x], x]];
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 + (4*n - 7)*x)*v[i - 1] - 3*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, 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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.