Real-rooted atlas

A196347 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(5\,n + 6\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
15611
25012072242
37502700324012967986
4150007200012960010368031104351384
5375000225000054000006480000388800093312019326120
61125000081000000243000000388800000349920000167961600335923201275523920
73937500003307500000119070000002381400000028576800000205752960008230118400141087744098215341840
8157500000001512000000006350400000001524096000000228614400000021946982400001316818944000451480780800677221171208642950081920
970875000000076545000000003674160000000010287648000000018517766400000022221319680000017777055744000091424858112000274274574336003656994324480855652058110080
103543750000000042525000000000022963500000000007348320000000000154314720000000002222131968000000022221319680000000152374763520000006856864358400000182849716224000021941965946880094121726392108800
11194906250000000025727625000000000154365750000000000555716700000000000133372008000000000022406497344000000002688779681280000000230466829824000000013828009789440000005531203915776000001327488939786240001448169752494080011388728893445164800
1211694375000000000016839900000000000001111433400000000000044457336000000000000120034807200000000000230466829824000000000322653561753600000000331872234946560000000248904176209920000000132748893978624000000477896018323046400001042682221795737600010426822217957376001503312213934761753600
137601343750000000000118580962500000000000853782930000000000000375664489200000000000011269934676000000000000243430589001600000000003894889424025600000000046738673088307200000000420648057794764800000002804320385298432000000013460737849432473600000440533238708699136000088106647741739827200081329213300067532800214973646592670930764800
145320940625000000000008939180250000000000000697256059500000000000003346829085600000000000001104453598248000000000000265068863579520000000000047712395444313600000000006543414232363008000000000687058494398115840000000054964679551849267200000003297880773110956032000000143907524644841717760000043172257393452515328000079702629034066182144000683165391720567275520033105941575271323337779200
Technical details
Node id
n-2cf8ae9156c81000 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-da2debf48cb6165b

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,             6;
  2:           50,           120,             72;
  3:          750,          2700,           3240,            1296;
  4:        15000,         72000,         129600,          103680,           31104;
  5:       375000,       2250000,        5400000,         6480000,         3888000,          933120;
  6:     11250000,      81000000,      243000000,       388800000,       349920000,       167961600,        33592320;
  7:    393750000,    3307500000,    11907000000,     23814000000,     28576800000,     20575296000,      8230118400,     1410877440;
  8:  15750000000,  151200000000,   635040000000,   1524096000000,   2286144000000,   2194698240000,   1316818944000,   451480780800,    67722117120;
  9: 708750000000, 7654500000000, 36741600000000, 102876480000000, 185177664000000, 222213196800000, 177770557440000, 91424858112000, 27427457433600, 3656994324480;
%F Formula
The row polynomials satisfy P(n,x) = (5*n+6*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 + 6*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 + 6*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.