Real-rooted atlas

A196347 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5\,n + 3\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1538
2506018128
375013508101623072
415000360003240012960194498304
537500011250001350000810000243000291603932160
611250000405000006075000048600000218700005248800524880188743680
739375000016537500002976750000297675000017860500006429780001285956001102248010569646080
8157500000007560000000015876000000019051200000014288400000068584320000205752960003527193600264539520676457349120
970875000000038272500000009185400000000128595600000001157360400000069441624000002777664960000714256704000107138505600714256704048704929136640
10354375000000002126250000000005740875000000009185400000000009644670000000006944162400000003472081200000001190427840000002678462640000035712835200002142770112003896394330931200
11194906250000000012863812500000000385914375000000006946458750000000083357505000000000700203042000000004201218252000000018005221080000000540156632400000010803132648000001296375917760007071141369600342882701121945600
1211694375000000000084199500000000000027785835000000000005557167000000000000750217545000000000072020884320000000005041461902400000000259275183552000000097228193832000000025927518355200000046669533039360000509122178611200025456108930560032916739307706777600
137601343750000000000592904812500000000002134457325000000000004695806115000000000007043709172500000000007607205906300000000006085764725040000000003651458835024000000001643156475760800000005477188252536000000013145251806086400000215104120463232000021510412046323200099278824829184003423340888001504870400
145320940625000000000004469590125000000000000174314014875000000000004183536357000000000000069028349890500000000000828340198686000000000007455061788174000000000051120423690336000000000268382224374264000000001073528897497056000000032205866924911680000007026734601798912000001054010190269836800009729324833260032000416971064282572800383414179456168545484800
Technical details
Node id
n-66d46e23b1cea6df · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b1029f17d1b64336

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:            5,             3;
  2:           50,            60,            18;
  3:          750,          1350,           810,            162;
  4:        15000,         36000,         32400,          12960,           1944;
  5:       375000,       1125000,       1350000,         810000,         243000,         29160;
  6:     11250000,      40500000,      60750000,       48600000,       21870000,       5248800,        524880;
  7:    393750000,    1653750000,    2976750000,     2976750000,     1786050000,     642978000,     128595600,     11022480;
  8:  15750000000,   75600000000,  158760000000,   190512000000,   142884000000,   68584320000,   20575296000,   3527193600,    264539520;
  9: 708750000000, 3827250000000, 9185400000000, 12859560000000, 11573604000000, 6944162400000, 2777664960000, 714256704000, 107138505600, 7142567040;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+3*n*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[(5*n + 3*n*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] = (5*n + 3*n*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.