Real-rooted atlas

A136124 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-1 - 6\,x - 9\,x^{2} - 2\,n - 6\,n x - n^{2}\right)\,P_{n-2}(x) + \left(3 + 6\,x + 2\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
23339981
320135718927774
4138631682529810817974
510782294123042014715324024388812
69381629242836332122882576140123937291067652
7904824313426844753313403323147987036401445927218713809744
8959256036227304577724405072911226743689862066816431661662126561191501712
9110961360450612504786852468774885420473527782185672655464781247099002590490196832836699488
101391411520601324300811333481444122662301408436391785384689144711745524072359991702962291520667155904944729949600
1118806852160858021258241726669862282023642514521534457524057898580548128122127803690755858111421060751201647157144929177147748371052800
12272636461440130473034329627799073607603488713073208287270332806616347319773846587278678531891158780063829703595353132290204761120872444628653144113245377644800
13422031435840021076006481280472263259824726291248402743255612707483762344515079731681538374355060450097152026351188122448942202208635629239134278241849237533829047961594323247308983308800
1469486541286400360576478298880845113529123712118675028371536011161072445108047439165799745683624111197236531310544641307333532427421237070529497308271024578921366104637367107706072512627900652547829694858930025510400
Technical details
Node id
n-dc9f5a840be72e54 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-41362b43d60d976c

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:         6,         3;
  2:        33,        39,         9;
  3:       201,       357,       189,        27;
  4:      1386,      3168,      2529,       810,        81;
  5:     10782,     29412,     30420,     14715,      3240,       243;
  6:     93816,    292428,    363321,    228825,     76140,     12393,      729;
  7:    904824,   3134268,   4475331,   3403323,   1479870,    364014,    45927,    2187;
  8:   9592560,  36227304,  57772440,  50729112,  26743689,   8620668,  1643166,  166212,   6561;
  9: 110961360, 450612504, 786852468, 774885420, 473527782, 185672655, 46478124, 7099002, 590490, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(n^2+2*n+1+(6*n+6)*x+9*x^2)*P(n-2,x) + (2*n+3+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(n^2 + 2*n + 1 + (6*n + 6)*x + 9*x^2)*P[n - 2, x] + (2*n + 3 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 + 2*n + 1 + (6*n + 6)*x + 9*x^2)*v[i - 2] + (2*n + 3 + 6*x)*v[i - 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

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.