Real-rooted atlas

A075502 → reverse

nonnegative coefficients

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

nx0x1x2x3x4x5x6x7x8x9Pn(1)
01214971
1142343343729
21701225514524018842
311053185308707443116807125399
41147686012005072270110588411176492026249
51196130343601504084101162355621494142382354336458010
6125222638907578166893511305903903558882252100034655764801719866701
71315367502016840548076277147176754012419145768365619029458133114035360715453821461

All 15 rows

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

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.