Real-rooted atlas

A136124 → basis A008277 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
2181221
31156060136
41241804803601045
51354202100420025209276
6148840672025200403202016093289
71631512176401058403175204233601814401047376
8180252040320352800169344042336004838400181440012975561
9199396083160997920698544027941760598752005987520019958400175721140
10112059401584002494800239500801397088004790016008981280007983360002395008002581284541
1111438580283140566280071351280570810240285405120085621536001427025600011416204800311351040040864292184
12116812012480480118918801902700801997835840136994457605993507520015982686720023974030080017435658240043589145600693347907421
1311951638078078023423400463783320618377760055653998400333923990400129859329600031166239104004249941696000283329446400065383718400012548540320876
14122421840122304043723680104936832017314577280197880883200155831195520083109970944002908848983040063465795993600793322449920004881984307200010461394944000241253367679185
Technical details
Node id
n-b25f7b2d6b8b4be7 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-caf342f7caa2ed5f

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 0)
  • 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;
  1: 1,  3;
  2: 1,  8,   12;
  3: 1, 15,   60,    60;
  4: 1, 24,  180,   480,    360;
  5: 1, 35,  420,  2100,   4200,    2520;
  6: 1, 48,  840,  6720,  25200,   40320,    20160;
  7: 1, 63, 1512, 17640, 105840,  317520,   423360,   181440;
  8: 1, 80, 2520, 40320, 352800, 1693440,  4233600,  4838400,  1814400;
  9: 1, 99, 3960, 83160, 997920, 6985440, 27941760, 59875200, 59875200, 19958400;
%F Formula
The row polynomials satisfy P(n,x) = (1+(2*n+1)*x)*P(n-1,x) - x^2*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(1 + (2*n + 1)*x)*P[n - 1, x] - 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; for(n = 1, N, my(i = n + 1); v[i] = (1 + (2*n + 1)*x)*v[i - 1] - 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.