Real-rooted atlas

A093375 → D → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2628
32418345
48096364216
5240400240605945
6672144012004809063888
71792470450402800840126715309
8460814336188161344056001344168858320
911520414726451256448302401008020162169216513
10281601152002073602150401411206048016800288027010787320
1167584309760633600760320591360310464110880264003960330112814669
121597448110081858560253440022809601419264620928190080396005280396129920232
1337273620766725271552805376082368005930496307507211531523088805720068644681334543665
14860160521830414536704246005762818816023063040138378246150144201801648048080080873654614119042784

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(2 - 3\,n + n^{2}\right)\,P_{n}(x) = \left(4\,n + 4\,n x + n x^{2} - 4\,n^{2} - 4\,n^{2} x - n^{2} x^{2}\right)\,P_{n-2}(x) + \left(-8\,n - 4\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 2\,x\)
Technical details
Node id
n-e924c6dc85f754f0 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-75dbd16a53b123f2)
Fingerprint
f6-49d880ced4b36b3a

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     0;
  1:     1;
  2:     6,     2;
  3:    24,    18,     3;
  4:    80,    96,    36,     4;
  5:   240,   400,   240,    60,     5;
  6:   672,  1440,  1200,   480,    90,     6;
  7:  1792,  4704,  5040,  2800,   840,   126,    7;
  8:  4608, 14336, 18816, 13440,  5600,  1344,  168,   8;
  9: 11520, 41472, 64512, 56448, 30240, 10080, 2016, 216, 9;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = (2*n+2+(2*n-2)*x)*P(n-1,x) - (2*x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((2*n + 2 + (2*n - 2)*x)*P[n - 1, x] - (2*x + x^2)*D[P[n - 1, x], x])/(n - 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((2*n + 2 + (2*n - 2)*x)*v[i - 1] - (2*x + x^2)*deriv(v[i - 1], x))/(n - 1)); 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

Candidate equalities ?

Same fingerprint ?

References

  1. [ASW52] Michael Aissen, I. J. Schoenberg and A. M. Whitney. On the generating functions of totally positive sequences I. Journal d’Analyse Mathématique, 2(1):93–103, 1952. doi:10.1007/BF02786970
  2. [AW24] Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989. arXiv
  3. [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
  4. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  5. [Edr53] Albert Edrei. Proof of a Conjecture of Schoenberg on the Generating Function of a Totally Positive Sequence. Canadian Journal of Mathematics, 5:86–94, 1953. doi:10.4153/CJM-1953-010-3
  6. [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
  7. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.