Real-rooted atlas

A075505 → reverse

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(1 + 20\,x + 10\,n x\right)\,P_{n-1}(x) + \left(-10\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1 + 30\,x + 100\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0130100131
116070010001761
211002500150001000027601
31150650090000310000100000506651
412101400035000030100006300000100000010674211
51280266001050000170100009660000012700000010000000251686881
6136046200264600069510000777000000302500000025500000001000000006524202561
7145075000588000022827000042525000003410500000093300000000511000000001000000000183991725451

All 15 rows

Technical details
Node id
n-9e0bd0b25f44a705 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-68c7410893bd483f

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,  30,    100;
  1: 1,  60,    700,     1000;
  2: 1, 100,   2500,    15000,      10000;
  3: 1, 150,   6500,    90000,     310000,      100000;
  4: 1, 210,  14000,   350000,    3010000,     6300000,       1000000;
  5: 1, 280,  26600,  1050000,   17010000,    96600000,     127000000,       10000000;
  6: 1, 360,  46200,  2646000,   69510000,   777000000,    3025000000,     2550000000,      100000000;
  7: 1, 450,  75000,  5880000,  228270000,  4252500000,   34105000000,    93300000000,    51100000000,     1000000000;
  8: 1, 550, 115500, 11880000,  639870000, 17948700000,  246730000000,  1457500000000,  2850100000000,  1023000000000,    10000000000;
  9: 1, 660, 170500, 22275000, 1590270000, 62739600000, 1323652000000, 13794000000000, 61150100000000, 86526000000000, 20470000000000, 100000000000;
%F Formula
The row polynomials satisfy P(n,x) = (1+(10*n+20)*x)*P(n-1,x) - 10*x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1 + 30*x + 100*x^2.
%t Mathematica
P[0, x_] := 1 + 30*x + 100*x^2;
P[n_, x_] := P[n, x] = Expand[(1 + (10*n + 20)*x)*P[n - 1, x] - 10*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 + 30*x + 100*x^2; for(n = 1, N, my(i = n + 1); v[i] = (1 + (10*n + 20)*x)*v[i - 1] - 10*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.