Real-rooted atlas

A071951 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
318413
412052881
514029232016669
6170109238241936326955
7111231922566447824116486488505
811687896121424561104585536699521281346209
9124017304453056420382411807616709638441984025623998521
10133034584142208023232176137922336243248704855764482519296512493956467
111440642843912128102868656111367372843809187844950550528102943667215116288102411596542533
121572112684969768838454187268743184645115521536013637811404810004044723212368356352906987522048307301505073
1317281881882209292812573337923456133324843611704934422848971591684191383868672201317730099214851097497654419456040969110471500693
14191030175646933744368755587214772137452829245330432002670745192243210015706455372812775469336115240412056994816178267589427232651714568192299893197116059
15111204673769400768098828100805533525204481621945675072023727383103283215957743722099204334351404617728387305285782937681002381579059221395375902720195910369281638410888674034993905
Technical details
Node id
n-04093e28420f148f · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-57cb71c89ecc0cb7

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,   2;
   3: 1,   8,     4;
   4: 1,  20,    52,       8;
   5: 1,  40,   292,     320,       16;
   6: 1,  70,  1092,    3824,     1936,        32;
   7: 1, 112,  3192,   25664,    47824,     11648,        64;
   8: 1, 168,  7896,  121424,   561104,    585536,     69952,      128;
   9: 1, 240, 17304,  453056,  4203824,  11807616,   7096384,   419840,     256;
  10: 1, 330, 34584, 1422080, 23232176, 137922336, 243248704, 85576448, 2519296, 512;
%F Formula
The row polynomials satisfy P(n,x) = (1+(n^2-n)*x)*P(n-1,x) - (2*n-2)*x^2*d/dx P(n-1,x) + x^3*d^2/dx^2 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 + (n^2 - n)*x)*P[n - 1, x] - (2*n - 2)*x^2*D[P[n - 1, x], x] + x^3*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; for(n = 2, N, my(i = n); v[i] = (1 + (n^2 - n)*x)*v[i - 1] - (2*n - 2)*x^2*deriv(v[i - 1], x) + x^3*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.