Real-rooted atlas

A136124 → basis A013610 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1336
29211242
32710814160336
481486107110263603024
524320256615105758262252030240
6729801936045846451093867365620160332640
7218730618180306578340108996312054427233841814403991680
865611137248470983538080905328914515956142245727778160181440051891840
919683413343379225819932318660787471431340472023492321798047729100296019958400726485760
10590491476225163368901053040504374240571222347105232465626029676051002430666144115191072023950080010897286400
111771475196312682015955282917202681224821935355405622864481145391233466804587086473235054392032156933417603113510400174356582400
125314411806899427735315325396974901543975854365597809662199543200219437472776070685339447716747355282344537841513728229047315840435891456002964061900800
131594323621785971103094369117793897658441473797942838980713115825967455874305566331495861810998419812522157562772128238537763448754764653440356647717440065383718400053353114214400
144782969212044959430414065952987679199441714450177263580522905711602027150857382382469238779474339125651417545623243548423882608233338423144595438182415077566597824059025146342400104613949440001013709170073600
Technical details
Node id
n-55badd6ba5138a5d · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-d86661683c23bbdc

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     1;
  1:     3,      3;
  2:     9,     21,      12;
  3:    27,    108,     141,       60;
  4:    81,    486,    1071,     1026,      360;
  5:   243,   2025,    6615,    10575,     8262,      2520;
  6:   729,   8019,   36045,    84645,   109386,     73656,     20160;
  7:  2187,  30618,  180306,   578340,  1089963,   1205442,    723384,    181440;
  8:  6561, 113724,  847098,  3538080,  9053289,  14515956,  14224572,   7778160,  1814400;
  9: 19683, 413343, 3792258, 19932318, 66078747, 143134047, 202349232, 179804772, 91002960, 19958400;
%F Formula
The row polynomials satisfy P(n,x) = (3+(n+2)*x)*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[(3 + (n + 2)*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; for(n = 1, N, my(i = n + 1); v[i] = (3 + (n + 2)*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.